text:
\section{Introduction}
\label{sec1}
The simulation of compressible flow is challenging when there present a broadband of length scales and discontinuities.
The numerical schemes must be of low dissipation in smooth region without damping the fine flow structures,
and be able to introduce sufficient dissipation in vicinity of discontinuities for suppressing non-physical oscillations.
In order to address this issue, many high-order shock capturing schemes,
like total variation diminishing (TVD)\cite{harten1983high},
essentially non-oscillatory (ENO)\cite{harten1987uniformly}
and weighted essentially non-oscillatory (WENO)~\cite{liu1994weighted} schemes, have been developed.
Among them, the WENO-JS scheme proposed by Jiang and Shu \cite{jiang1996efficient} exhibits high-order accuracy
and robust shock-capturing properties
and have been extensively employed in direct numerical simulations (DNS) and large eddy simulations (LES).
In spite of these advantages, the classic WENO-JS scheme still faces several shortcomings.
It suffers from order-degeneration,
e.g. not able to recover the formal order of accuracy near critical points where the first or higher derivatives vanish~\cite{henrick2005mapped},
and is too dissipative for sustaining fine structures in turbulence or aeroacoustic flows~\cite{martin2006bandwidth}.
Furthermore, the computation may fail when very strong discontinuities exist
or multiple discontinuities are too close to each other~\cite{gerolymos2009very}.

In order to overcome the drawbacks of order-degeneration and excessive dissipation
of the classical WENO-JS scheme,
one promising approach is to modify the weighting strategy, e.g. the adaptation mechanism, by increasing the weight of the optimal upwind linear scheme in relatively smooth regions.
Following this idea, various schemes have been developed,
such as WENO-M scheme~\cite{henrick2005mapped} with a corrective mapping applied to the classical WENO weights,
WENO-Z scheme~\cite{borges2008improved,castro2011high} with a new high-order smoothness indicator
and WENO-RL or WENO-RLTV~\cite{taylor2007optimization} scheme switching on the optimal weights
when the magnitude of smoothness indicators are comparable.
Another approach is to introduce the contribution of the downwind stencil and use optimal weights
for central scheme instead of upwind scheme as pointed in Refs.~\cite{martin2006bandwidth,sun2011class,hu2010adaptive}.
Take WENO-CU6 scheme~\cite{hu2010adaptive} as an example, Hu et al. employed the 6th-order central scheme as the optimal linear scheme
and introduced a new smooth indicator
for the adaptation between central and upwind stencils.
Later, this scheme was extended to WENO-CU6-M with a physically-motivated scale separation approach for implicit large eddy simulation (ILES)~\cite{hu2011scale}.
While WENO-CU6-M works well for incompressible and compressible turbulence simulation with low dissipation
and maintains the shock-capturing capability,
it produces spurious waves because of the non-dissipative optimal scheme.
In addition, as demonstrated in Ref.~\cite{fu2016family}, one version of WENO-CU6-M scheme fails in the benchmark test cases "Interacting blast waves"
and "Double Mach reflection"
and suggests inferior robustness than the classic WENO-JS scheme.
Fu et al.~\cite{fu2016family} proposed a family of TENO scheme which dynamically
assembles a set of low-order candidate stencils with incrementally increasing stencil width.
The combination of low-order approximation polynomials with incremental stencils
makes TENO scheme considerably more robust.
However, positivity-preserving methods, such as those in Ref.~\cite{hu2013positivity,zhang2010positivity},
are still required for TENO schemes when high density ratio or strong shock waves present.

Recently, Wang et al.~\cite{wang2018incremental} developed a 5th-order incremental-stencil
WENO (denote as WENO-IS) scheme for multi-phase flows.
Since both 2- and 3-point stencils are used,
in the region with strong discontinuities,
the reconstruction would prefer to choose the 2-point stencils,
which makes the scheme, compared to the classic WENO-JS schemes,
very robust for material interface with high density ratio and strong shock waves.
In their weighting strategy,
a slight modification is introduced to decrease the weights for 2-point stencils in relatively smooth region
 and to avoid excessive dissipation and order degeneration near critical points.
However,
this scheme is still too dissipative to capture fine flow structures,
as also shown in Section 4, due to the too much weights for the 2-point stencils.

In this paper,
we propose a L2-norm regularized incremental-stencil WENO scheme following
the idea of neural-network optimization~\cite{nielsen2015neural}.
Similar to WENO-IS scheme, both incremental 2- and 3-point stencils are used for the 5th-order reconstruction.
The difference is that a L2-norm regularization is introduced to address the problem of order degeneration and excessive dissipation.
First, the L2-norm errors between the approximations from 2-point stencils and a classical 3-point stencil are utilized to modulate the weights of the incremental stencils.
Then, a high-order non-dimensional discontinuity detector is adopted as the regularization parameter to control the influence of the error term.
With a further hybridization with a linear scheme, less numerical dissipation in very smooth region and considerable higher computational efficiency are achieved.
The reminder of the paper is organized as follows.
Section 2 gives a brief review of the WENO-IS scheme following Wang et al.~\cite{wang2018incremental}
and the application of WENO-IS to Euler equations is given.
In Section 3, the L2-norm regularization
and control parameter are proposed.
Benchmark test cases with strong discontinuities and fine flow structures are presented in Section 4,
and brief concluding remarks are given in the last Section 5.

\section{WENO-IS scheme}
\label{sec1}
In this section, we briefly review the 5th-order WENO-IS scheme for hyperbolic conservation laws
following Ref.~\cite{wang2018incremental}.
First, we illustrate the details of WENO-IS scheme based on the discretization of one-dimensional scalar hyperbolic conservation law.
Then, its application in Euler equations is given.
\subsection{Semi-discretization of a hyperbolic conservation law}
\label{subsec1}
For simplicity, we consider the following one-dimensional scalar hyperbolic conservation law
 \begin{equation}
 \frac{\partial {u}}{{\partial t}} +
 \frac{\partial {f\left({u} \right) }}{{\partial x}}=0,
 \label{goveq3}
 \end{equation}
 where $u$ denotes the conservative variable and $f\left(u \right) $ is the flux function.
 Here, the characteristic velocity is assumed to be positive with $\frac{\partial {f\left({u} \right) }}{{\partial u}}>0$.

 On a uniform grid, we denote $x_j=j\Delta x$, where $\Delta x$ is the grid spacing.
 The quantities at the point $x_j$ are identified by the subscript $j$.
 The semi-discretized form of Eq.~\eqref{goveq3} using the method of lines can be
 approximated by a conservative finite difference scheme as
 \begin{equation}
\frac{d {{u}_j}}{{d t}} =
-\frac{1}{\Delta x} \left(h_{j+1/2}-h_{j-1/2} \right) ,
\label{eq2}
\end{equation}
 where $h_{j\pm1/2}=h\left( x_{j\pm1/2}\right) $ and is implicitly defined by
  \begin{equation}
 f\left[u\left( x\right)  \right]  =
 \frac{1}{\Delta x} \int_{x-\Delta x/2}^{x+\Delta x/2} h(\xi)d\xi.
 \label{eq3}
 \end{equation}
 %
 Numerically, $h_{j\pm1/2}$ are approximated by high-order polynomial interpolations $\widehat{f}_{j\pm1/2}$ or numerical fluxes and Eq.~\eqref{eq2} can be expressed as
  \begin{equation}
 \frac{d {u_j}}{{d t}} \approx
 -\frac{1}{\Delta x} \left(\widehat{f}_{j+1/2}-\widehat{f}_{j-1/2} \right).
 \label{eq4}
 \end{equation}

\subsection{Candidate stencils with incremental width}
\label{subsec1}
 For 5th-order WENO-IS scheme, the 5-point stencil $S_5$ as shown in Fig.~\ref{stencil} is served as
 the full stencil for the evaluation of numerical fluxes $\widehat{f}_{j+1/2}$.
 Different from the classic WENO-JS scheme, which
 subdivides the full stencil into three stencils with the same width,
 the WENO-IS scheme uses four stencils with incremental width
 including two 2-point and two 3-point stencils, e.g. $S_0$, $S_1$, $S_2$ and $S_3$
 as shown in Fig.~\ref{stencil}, for reconstruction.

 \begin{figure}[tbh]
 	\begin{center}
 		\begin{tikzpicture}[
 		dot/.style 2 args={circle,draw=#1,fill=#2,inner sep=2.5pt},
 		square/.style 2 args={draw=#1,fill=#2,inner sep=3pt},
 		mystar/.style 2 args={star,draw=#1,fill=#2,inner sep=1.5pt},
 		mydiamond/.style 2 args={diamond,draw=#1,fill=#2,inner sep=1.5pt},
 		scale=0.8
 		]

 		\draw[black,thick]  (0,-1) grid (15,-1);
 		\draw[black,thick,yshift=-0.3cm]  (6.25,-3) grid (8.75,-3);
 		\draw[black,thick,yshift=-0.3cm]  (3.75,-4) grid (6.25,-4);
 		\draw[black,thick,yshift=-0.3cm]  (6.25,-5) grid (11.25,-5);
 		\draw[black,thick,yshift=-0.3cm]  (1.25,-6) grid (6.25,-6);
 		\draw[gray,thick,yshift=-0.3cm]  (3.75,-7) grid (8.75,-7);
 		\draw[black,thick,yshift=-0.3cm]  (1.25,-8) grid (11.25,-8);
 		\foreach \Fila in {1.25,3.75,6.25,8.75,11.25,13.75}{\node[dot={black}{black}] at (\Fila,-1) {};}
 		\foreach \Fila in {7.5}{\node[square={black}{white}] at (\Fila,-1) {};}
 		\foreach \Fila in {6.25,8.75}{\node[dot={black}{black}] at (\Fila,-3.3) {};}
 		\foreach \Fila in {3.75,6.25}{\node[dot={black}{black}] at (\Fila,-4.3) {};}
 		\foreach \Fila in {6.25,8.75,11.25}{\node[dot={black}{black}] at (\Fila,-5.3) {};}
 		\foreach \Fila in {1.25,3.75,6.25}{\node[dot={black}{black}] at (\Fila,-6.3) {};}
 		\foreach \Fila in {3.75,6.25,8.75}{\node[dot={gray}{gray}] at (\Fila,-7.3) {};}
 		\foreach \Fila in {1.25,3.75,6.25,8.75,11.25}{\node[dot={black}{black}] at (\Fila,-8.3) {};}
 		\node[below] at (1.25,-1.3) {$x_{j-2}$};
 		\node[below] at (3.75,-1.3) {$x_{j-1}$};
 		\node[below] at (6.25,-1.3) {$x_{j}$};
 		\node[below] at (8.75,-1.3) {$x_{j+1}$};
 		\node[below] at (11.25,-1.3) {$x_{j+2}$};
 		\node[below] at (13.75,-1.3) {$x_{j+3}$};
 		\node[below] at (7.5,-1.3) {$x_{j+1/2}$};
 		\node[left]  at (6.25,-3.3) {$S_{0}$};
 		\node[left]  at (3.75,-4.3) {$S_{1}$};
 		\node[left]  at (6.25,-5.3) {$S_{2}$};
 		\node[left]  at (1.25,-6.3) {$S_{3}$};
 		\node[left]  at (3.75,-7.3) {$S_{01}$};
 		\node[left]  at (1.25,-8.3) {$S_{5}$};
 		\end{tikzpicture}
 	\end{center}
 	\caption{Full stencil and candidate stencils with incremental width for WENO-IS reconstruction.
 	Here, $S_{01}$ is an original 3-point stencil used for WENO-JS reconstruction.  }
 	\label{stencil}
 \end{figure}
 The numerical flux $\widehat{f}_{j +1/2} $ is computed by a
 convex combination of the four candidate stencils fluxes as
\begin{equation}
\widehat{f}_{j +1/2}=
\sum_{k=0}^{3} \omega_k \widehat{f}_{k,j+1/2} ,
\label{eq5}
\end{equation}
where $\widehat{f}_{k,j+1/2}$ and $\omega_k$ are the reconstructed fluxes from candidate stencils
and their non-linear weights, respectively.
These reconstructed fluxes are
\begin{equation}
\begin{split}
&\widehat{f}_{0,j+1/2}=\frac{1}{2}f_j+\frac{1}{2}f_{j+1}, \\
&\widehat{f}_{1,j+1/2}=-\frac{1}{2}f_{j-1}+\frac{3}{2}f_{j} , \\
&\widehat{f}_{2,j+1/2}=\frac{1}{3}f_{j}+\frac{5}{6}f_{j+1}-\frac{1}{6}f_{j+2} , \\
&\widehat{f}_{3,j+1/2}=\frac{1}{3}f_{j-2}-\frac{7}{6}f_{j-1}+\frac{11}{6}f_{j}.  \\
\end{split}
\label{eq6}
\end{equation}
From Taylor series expansion, we can obtain
\begin{equation}
\widehat{f}_{k,j +1/2}=
h_{j+1/2}+A_k \Delta x^{r_k}+O\left( \Delta x^{r_k+1}\right)  ,
\label{eq7}
\end{equation}
where $A_k$, $k=0, 1, 2, 3$ are independent of $\Delta x$ and $r_k$ denote the points of the candidate stencil.

The non-linear weights $\omega_k$ are given by
\begin{equation}
\omega_k=
\frac{\alpha_k}{\sum\nolimits_{s=0}^{3}\alpha_s} , \quad \alpha_k=d_k\left(1+\frac{\tau_5}{\beta_k+\varepsilon} \right),
\label{eq8}
\end{equation}
where coefficients $d_0=\frac{4}{10}, d_1=\frac{2}{10}, d_2=\frac{3}{10}, d_3=\frac{1}{10}$ are the optimal weights
and $\varepsilon$ is a small positive parameter to prevent division by zero and we choose $\varepsilon=10^{-20}$ in this paper.
 $\beta_k$ are the smoothness indicators of each candidate stencil,
while $\tau_5$ is the global reference smoothness indicator on the full stencil. Following Jiang and Shu ~\cite{jiang1996efficient},
the smoothness indicators at $ S_k$ are evaluated by
 \begin{equation}
 \beta_k=\sum\limits_{l=1}^{r_k-1}\Delta x^{2l-1} \int_{x_{j-1/2}}^{x_{j+1/2}}\left(\frac{d^l}{dx^l}
 \widehat{f}_k\left( x\right) \right)^2 dx.
 \label{eq9}
 \end{equation}
The global reference smoothness indicator is obtained by
\begin{equation}
\tau_5=
\sum\limits_{l=3}^{4}\Delta x^{2l-1} \int_{x_{j-1/2}}^{x_{j+1/2}}\left(\frac{d^l}{dx^l} \widehat{f}_k\left( x\right) \right)^2 dx,
\label{eq10}
\end{equation}
which only contains the high-order component of the smoothness indicator for the full stencil~\cite{fu2016family,wang2018incremental}.
The formulations of $\beta_k$ and $\tau_5$ in terms of cell-centered fluxes $f_j$ can be expressed as
\begin{equation}
\begin{split}
&\beta_0=\left(f_{j+1}-f_{j} \right) ^2, \\
&\beta_1=\left(f_{j}-f_{j-1} \right) ^2, \\
&\beta_2=\frac{13}{12}\left(f_{j}-2f_{j+1}+f_{j+2} \right) ^2+ \frac{1}{4}\left(3f_{j}-4f_{j+1}+f_{j+2} \right) ^2, \\
&\beta_3=\frac{13}{12}\left(f_{j-2}-2f_{j-1}+f_{j} \right) ^2+ \frac{1}{4}\left(f_{j-2}-4f_{j-1}+3f_{j} \right) ^2, \\
&\tau_5=\frac{13}{12}\left(f_{j+2}-4f_{j+1}+6f_{j}-4f_{j-1}+f_{j-2} \right) ^2+
\frac{1}{4}\left(f_{j+2}-2f_{j+1}+2f_{j-1}-f_{j-2} \right) ^2, \\
\end{split}
\label{eq11}
\end{equation}
and their Taylor series expansions at $x_j$ are
\begin{equation}
\begin{split}
&\beta_0={f^\prime}_j ^2 \Delta x^2+{f^\prime}_j{f^\prime{^\prime}}_j \Delta x^3 +
\left(\frac{1}{4}{f^\prime{^\prime}}_j^2+\frac{1}{3}{f^\prime}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^4+
\left(\frac{1}{12}{f^\prime}_j{f^\prime{^\prime}{^\prime}{^\prime}}_j+
\frac{1}{6}{f^\prime{^\prime}}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^5+
O\left( \Delta x^6\right) , \\
&\beta_1={f^\prime}_j ^2 \Delta x^2-{f^\prime}_j{f^\prime{^\prime}}_j \Delta x^3 +
\left(\frac{1}{4}{f^\prime{^\prime}}_j^2+\frac{1}{3}{f^\prime}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^4-
\left(\frac{1}{12}{f^\prime}_j{f^\prime{^\prime}{^\prime}{^\prime}}_j+
\frac{1}{6}{f^\prime{^\prime}}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^5+
O\left( \Delta x^6\right) , \\
&\beta_2={f^\prime}_j ^2 \Delta x^2+
\left(\frac{13}{12}{f^\prime{^\prime}}_j^2-\frac{2}{3}{f^\prime}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^4+
\left(-\frac{1}{2}{f^\prime}_j{f^\prime{^\prime}{^\prime}{^\prime}}_j+
\frac{13}{6}{f^\prime{^\prime}}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^5+
O\left( \Delta x^6\right) , \\
&\beta_3={f^\prime}_j ^2 \Delta x^2+
\left(\frac{13}{12}{f^\prime{^\prime}}_j^2-\frac{2}{3}{f^\prime}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^4-
\left(-\frac{1}{2}{f^\prime}_j{f^\prime{^\prime}{^\prime}{^\prime}}_j+
\frac{13}{6}{f^\prime{^\prime}}_j{f^\prime{^\prime}{^\prime}}_j \right)\Delta x^5+
O\left( \Delta x^6\right) , \\
&\tau_5={f^\prime{^\prime}{^\prime}}_j\Delta x^6+\frac{13}{12}{f^\prime{^\prime}{^\prime}{^\prime}}_j\Delta x^8
+ O\left( \Delta x^{10}\right). \\
\end{split}
\label{eq12}
\end{equation}
Substituting Eq.~\eqref{eq12} to Eq.~\eqref{eq8}, we can observe that
\begin{equation}
\frac{\tau_5}{\beta_k+\varepsilon}=O\left(\Delta x^4 \right)  , \quad k=0,1,2,3.
\label{eq13}
\end{equation}
Therefore, the condition of $\omega_k-d_k=O\left(\Delta x^4 \right)$ is satisfied at the region without critical points,
 implying that
the WENO-IS scheme has the expected 5th-order convergence from the analysis in Ref.~\cite{wang2018incremental}.

\subsection{Weighting strategy}
\label{subsec1}
From Eq.~\eqref{eq9} and Eq.~\eqref{eq11}, we can observe that the smoothness indicators of 2-point stencils
are evaluated from the 1st-order derivatives of linear functions,
while these of 3-point stencils are from the 1st and 2nd-order derivatives of parabolic functions.
Due to the lack of 2nd-order derivative term, the smoothness indicator of 2-point stencils can be much smaller
than that of stencils with 3-point especially near critical point and larger weights will be posed on 2-point stencils,
which may cause order degeneration eventually. In order to address this issue,
Wang et al.~\cite{wang2018incremental} made a slight modification of the weighting strategy in Eq.~\eqref{eq8} as
\begin{equation}
\alpha_0=d_0\left(1+\frac{\tau_5}{\beta_0+\varepsilon }\cdot \frac{\tau_5}{\beta_{01}+\varepsilon } \right), \quad
\alpha_1=d_1\left(1+\frac{\tau_5}{\beta_1+\varepsilon }\cdot \frac{\tau_5}{\beta_{01}+\varepsilon } \right),
\label{eq14}
\end{equation}
where
\begin{equation}
\beta_{01}=\frac{13}{12}\left(f_{j-1}-2f_{j}+f_{j+1} \right) ^2+ \frac{1}{4}\left(f_{j-1}-f_{j+1}\right) ^2
\label{eq15}
\end{equation}
is the smoothness indicator of $S_{01}$, an original 3-point stencil of WENO-JS as shown in Fig.~\ref{stencil},
and is obtained from 1st and 2nd-order derivatives.
This modification decreases the weights of 2-point stencils considerably and avoids order degeneration at critical point.

In relatively smooth region, the effectiveness of this modification
has been confirmed by a number of test cases in Ref.~\cite{wang2018incremental}.
The WENO-IS scheme achieves the formal order of accuracy and produces less errors than
classical WENO-JS scheme.
However, in the region with shocks or high wavenumber waves,
this modification may not decrease the weights of 2-point stencils sufficiently,
which can still lead  to excessive numerical dissipation.
Suppose there is a shock locating in stencil $S_3$ and the solutions in
stencils $S_0$, $S_1$ and $S_2$ are smooth (see Fig.~\ref{stencil}).
Under this condition, $\tau_5$ is comparable to $\beta_3$ and $\tau_5=O\left( 1\right) $,
while $\beta_k=O\left(\Delta x^2 \right)  $(k=0,1,2,01) are all small numbers.
It is straightforward to show that
\begin{equation}
\frac{\tau_5}{\beta_0+\varepsilon }\cdot \frac{\tau_5}{\beta_{01}+\varepsilon } \gg \frac{\tau_5}{\beta_{2}+\varepsilon }, \quad
\frac{\tau_5}{\beta_1+\varepsilon }\cdot \frac{\tau_5}{\beta_{01}+\varepsilon } \gg \frac{\tau_5}{\beta_{2}+\varepsilon },
\label{eq144}
\end{equation}
which means larger weights will be posed on 2-point stencils.
Same conclusion can be drawn when the shock locates in other stencils.
Thus, this modification enlarges the numerical dissipation near the discontinuities.
Although larger numerical dissipation improves the robustness of the scheme,
this modification imposes too much excessive numerical dissipation to resolve the fine flow structures with broadband length scales well (also see the cases in section 4).

Note that TENO scheme ~\cite{fu2016family} avoid the problem of large weight posed on low-order stencils
 by a ENO-like strategy.
However, the numerical stability of TENO scheme is inferior to WENO-JS hence also WENO-IS scheme.

\subsection{Application in Euler equations}
\label{subsec1}
For simplicity, the detailed procedure on the application of WENO-IS for Euler equations is illustrated based on
the one-dimensional Euler equations of gas dynamics
\begin{equation}
\frac{\partial \bm{U}}{{\partial t}} +
\frac{\partial {\bm{F}\left(\bm{U} \right) }}{{\partial x}}=0,
\label{goveq4}
\end{equation}
where
\begin{equation}
\bm{U}=\left[\begin{array}{c}
\rho \\ \rho u \\ {E}
\end{array} \right] ,
\bm{F}\left(\bm{ U}\right) =\left[\begin{array}{c}
\rho u \\ \rho u^2+p \\ {\left( E+p\right)u }
\end{array} \right]. \\
\end{equation}
Here, $\rho$ is density, $p$ is pressure and $u$ denotes the velocity in $x-$direction.
$E=p/\left( \gamma-1\right) +1/2\rho u^2$ is total energy per unit volume with ideal-gas equation of states used.

Within a uniform grid, at the point $x_j$, the semi-discretized form of Eq. \eqref{goveq4} gives
\begin{equation}
\frac{d {\bm{U}_j}}{{d t}} =-
\frac{1}{\Delta x}\left(\hat{\bm{F}}_{j+1/2} -\hat{\bm{F}}_{j-1/2}\right) ,
\label{goveq5}
\end{equation}
where $\hat{\bm{F}}_{j\pm1/2}$ are the numerical fluxes at $x_{j\pm 1/2}$.
With a typical characteristic-wise method, the numerical fluxes
are reconstructed within the local characteristic flied.
The procedure to obtain $\hat{\bm{F}}_{j+1/2}$ includes the following steps:

(1) At $x_{j+ 1/2}$, evaluate the Jacobian matrix $\bm{A}=\partial \bm{F}/\bm{U}$ at a Roe-average state.

(2) Compute the left and right eigenvector matrix $\bm{L}_{j+1/2}^s$, $\bm{R}_{j+1/2}^s \left(s=1,2,3 \right) $
as well as the eigenvalues $\lambda_{j+1/2}^s \left(s=1,2,3 \right)$
of the Roe-average Jacobian matrix $\bm{A}$ .

(3) Within each candidate stencils, transform the physical fluxes and conservative variables into characteristic filed as
\begin{equation}
v_m^s =\bm{L}_{j+1/2}^s \cdot \bm{U}_m, \quad
g_m^s =\bm{L}_{j+1/2}^s \cdot \bm{F}_m,
\label{eqchara}
\end{equation}
where $j-2<m<j+3$ and $s=1,2,3$.

(4) Carry out flux splitting in characteristic space. We can obtain
\begin{equation}
f_m^{s,\pm} =\frac{1}{2} \left(g_m^s \pm \alpha^s v_m^s \right) ,
\label{flxuspliting}
\end{equation}
where $\alpha^s=\left|\lambda_{j+1/2}^s \right| $ for Roe flux (RF) splitting.
Otherwise, $\alpha^s=max \left|\lambda_{l}^s \right| $ for Lax-Friedrichs flux (LF) splitting with
 $l$ representing the entire computational domain or for local Lax-Friedrichs flux (LLF) splitting when $l$ represents the local stencil.

 (5) Then, reconstruct the numerical flux with WENO-IS scheme, which gives
 \begin{equation}
 f_{j+1/2}^{s,+} =\sum\limits_{k=0}^{3} \omega_k^{+}  f_{k,j+1/2}^{s,+} , \quad
 f_{j+1/2}^{s,-} =\sum\limits_{k=0}^{3} \omega_k^{-}  f_{k,j+1/2}^{s,-} ,
 \label{reconstruction}
 \end{equation}
where $f_{k,j+1/2}^{s,+}$ and $\omega_k^{+}$ are the positive fluxes by candidate stencils
and their non-linear weights as defined in Eq.~\eqref{eq6} and Eq.~\eqref{eq8}, respectively.
The negative term  $f_{k,j+1/2}^{s,-}$ and $\omega_k^{-}$ can be computed in the similar way according to the symmetry at the point $x_{j+1/2}$.
The numerical flux in each characteristic space can be finally evaluated by
 \begin{equation}
\hat{f}_{j+1/2}^{s} =f_{j+1/2}^{s,+}+f_{j+1/2}^{s,-} .
\label{reconstruction1}
\end{equation}

(6) At last, transform the numerical flux back into physical space as
 \begin{equation}
\hat{\bm{F}}_{j+1/2}  = \sum\limits_{s=0}^{2} \bm{R}_{j+1/2}^{s} \hat{f}_{j+1/2}^{s}.
\label{reconstruction2}
\end{equation}

\section{A L2-norm regularized WENO-IS scheme}
\label{sec1}
In Ref.~\cite{nielsen2015neural} (their section 3.2.1),
in order to find a way to reduce overfitting of the neural-network,
an optimization technique, known as L2-norm regularization, is introduced.
The essential idea of L2-norm regularization is to add an extra term, called the regularization term,
to the original cost function which only contains the L2-norm error term, with a hyper-parameter to control the relative amount of regularization.
This technique can be viewed as a way of compromising between the original L2-norm error term and the L2-norm regularization term,
which makes the neural-network better at generalizing beyond the training data.

In the classical WENO schemes, as an analog of optimization technique, we can find that only the regularization terms based on the smoothness indicators are utilized.
The L2-norm error term is neglected due to the same order of accuracy for each candidate stencil.
However, for WENO-IS scheme, the reconstruction accuracy of 2-point stencil is different from that of the 3-point stencil, therefore,
neglecting the L2-norm error terms is not suitable anymore.
To cope with the discrepancy of numerical accuracy between 2-point and 3-point stencils,
an optimization strategy, L2-norm regularization, is introduced to the WENO weighting strategy.

\subsection{L2-norm regularization}
\label{subsec1}

For the conservation law in Eq.~\eqref{goveq3},
a form of L2-norm error term is
 \begin{equation}
E_k=\frac{1}{ \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}}\left( \widehat{f}_k\left( x\right)-{f}_k\left( x\right) \right)^2 dx,
\label{eq16}
\end{equation}
which is obtained by the difference between reconstruct flux and reference flux.
For the 5-th order WENO-IS scheme,
the three reconstructed fluxes of classic WENO-JS scheme are chosen as the reference fluxes.
Therefore, the L2-norm error term of each stencil for WENO-IS scheme can be evaluated by
\begin{equation}
\begin{split}
&E_0=\frac{1}{ \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}}
\left( \widehat{f}_0\left( x\right)-{f}_{01}\left( x\right) \right)^2 dx, \\
&E_1=\frac{1}{ \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}}
\left( \widehat{f}_1\left( x\right)-{f}_{01}\left( x\right) \right)^2 dx, \\
&E_2=\frac{1}{ \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}}\left( \widehat{f}_2\left( x\right)-{f}_2\left( x\right) \right)^2 dx, \\
&E_3=\frac{1}{ \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}}\left( \widehat{f}_3\left( x\right)-{f}_3\left( x\right) \right)^2 dx, \\
\end{split}
\label{eq17}
\end{equation}
where $f_{01}\left( x\right)$, $f_{2}\left( x\right)$ and $f_{3}\left( x\right)$ are the reconstructed
fluxes of the corresponding stencils $S_{01}$, $S_2$ and $S_3$ as shown in Fig.~\ref{stencil}.
Since the reconstructed fluxes and the reference fluxes of stencil $S_2$ and $S_3$ are the same,
their L2-norm error terms equal to zero accordingly. Therefore, the above error terms yield
\begin{equation}
\begin{split}
&E_0=\frac{1}{45}\left( f_{j+1}^2-4f_j f_{j+1}+2f_{j-1} f_{j+1}+4f_j^2 -4f_{j-1} f_{j}+f_{j-1}^2\right) , \\
&E_1=\frac{1}{45}\left( f_{j+1}^2-4f_j f_{j+1}+2f_{j-1} f_{j+1}+4f_j^2 -4f_{j-1} f_{j}+f_{j-1}^2\right), \\
&E_2=0 , \\
&E_3=0 . \\
\end{split}
\label{eq18}
\end{equation}

With L2-norm regularization, the weighting strategy of WENO-IS scheme is modified as
\begin{equation}
\omega_k^r=
\frac{\alpha_k^r}{\sum\nolimits_{s=0}^{3}\alpha_s^r} , \quad \alpha_k^r=d_k\left(1+\frac{\lambda\tau_5}{\lambda\beta_k+E_k+\varepsilon} \right),
\label{eq8-r}
\end{equation}
where $\lambda$ is known as the regularization parameter and we will discuss it in later subsection.
The Taylor series expansions at $x_j$ of the L2-norm error term $E_k$ as shown in Eq.~\eqref{eq18} are
\begin{equation}
\begin{split}
&E_0=E_1=\frac{1}{45}\left( {f^{\prime \prime}}_j \Delta x^4+\frac{1}{6}{f^{\prime \prime}}_j
{f^{\prime \prime\prime \prime}}_j \Delta x^6+O\left( \Delta x^6\right)\right) , \\
&E_2=E_3=0. \\
\end{split}
\label{eq277}
\end{equation}
It is straightforward to check from Eq.~\eqref{eq12} that $\omega_k^r-d_k=O\left(\Delta x^4 \right)$ is still satisfied for the present weighting strategy.
Hence, with L2-norm regularization the modified WENO-IS scheme also has the expect 5th-order convergence at non-critical points.
Note that, this modification doesn't solve the order degeneration problem at critical points.
Although some weighting strategies can be applied to cope with this case as noted by Borges et al.~\cite{borges2008improved},
it will increase numerical dissipation inevitably. Alternatively, in the present paper,
we introduce a hybrid scheme to resolve the
order degeneration problem, which will be discussed in section 3.3.

\subsection{An adaptive weights control }
\label{subsec1}
It is obvious that the effect of the regularization is to
make up the discrepancy of numerical accuracy between 2-point and 3-points stencils
so as to reduce the weights of 2-point stencils.
The relative importance of the regularization term and the added L2-norm error term depends on the value of $\lambda$.
Here, we introduce a non-dimensional discontinuity detector following Hu et al.~\cite{hu2015efficient}
to devise an adaptive regularization parameter.
Due to the design of non-dimensional discontinuity detector is based on Euler equations, in this section,
the detailed procedure is illustrated using one dimensional Euler equations as presented in section 2.3.

The non-dimensional discontinuity detector is defined by
\begin{equation}
\sigma^s=\left(\frac{\Delta v_{j+\frac{1}{2}}^s}{\tilde{\rho}} \right) ^{2}, \\
\label{eq27}
\end{equation}
where
\begin{equation}
\Delta v_{j+\frac{1}{2}}^s=\frac{1}{60} \bm{L}_{j+1/2}^s \cdot
\left(\bm{U}_{j-2}-5\bm{U}_{j-1}+10\bm{U}_{j}-10\bm{U}_{j+1}+5\bm{U}_{j+2}-\bm{U}_{j+3} \right) , \\
\label{eq28}
\end{equation}
and ${\tilde{\rho}}$ is the Roe-average density corresponding to the Jacobian matrix $\bm{A}$ at $x_{j+\frac{1}{2}}$.

We define a regularization parameter by
\begin{equation}
\lambda=c\sigma^s, \\
\label{eq29}
\end{equation}
where $c$ is a positive constant and we set $c=1$ in this work.
When the full 6-point stencil contains discontinuities,
the regularization parameter becomes a large number,
which makes the original smoothness indicator the leading term
and 2-point stencils will be assigned larger weights.
 Note that, the relatively large weights of 2-point stencils evaluated by the present method
 are less than these of the original WENO-IS scheme under this condition.
 As will be shown in section 4, with this treatment the scheme is still robust
 even for strong discontinuities and less numerical dissipation is imposed.
 Otherwise, the reconstruction prefer to choose 3-point stencils in the flow field
 with fine flow structures or the smooth region where $\lambda$ is small.

\subsection{The hybrid scheme }
\label{subsec1}
Although the added L2-norm error term and the designed adaptive parameter have the advantages as discussed above,
the modified WENO-IS scheme is not suitable for smooth region due to the preference of 3-point stencil of
the reconstruction when $\lambda$ is very small.
Furthermore, compared with its optimal linear scheme, the calculation of the modified WENO-IS scheme is time-consuming.
To overcome these limitations, with the discontinuity detector presented in above section,
it is easy to introduce a hybrid scheme flowing Hu et al.~\cite{hu2015efficient} to achieve better accuracy in the smooth region
and improve computational efficiency.
The numerical flux of this hybrid scheme is switching from that of the nonlinear WENO-IS scheme and
its optimal linear upwind scheme, which is given by
\begin{equation}
\hat{\bm{F}}_{j+1/2}=\sigma_{j+1/2} \hat{\bm{F}}_{j+1/2}^{UPS}+\left(1- \sigma_{j+1/2} \right) \hat{\bm{F}}_{j+1/2}^{WENO-IS} , \\
\label{eqhy}
\end{equation}
where $\sigma_{j+1/2}$ equals to one in smooth region otherwise zero in vicinity of discontinuities.
Here, the numerical flux $\hat{\bm{F}}_{j+1/2}^{UPS}$ of the optimal linear upwind scheme is evaluated by
\begin{equation}
\begin{split}
&\hat{\bm{F}}_{j+1/2}^{UPS}=\frac{1}{60}\left(\bm{F}_{j-2} -8\bm{F}_{j-1}+37\bm{F}_{j}+37\bm{F}_{j+1}-8\bm{F}_{j+2}+\bm{F}_{j+3} \right)   \\
& + \sum\limits_{s=0}^{2} \bm{R}_{j+1/2}^{s} \lambda ^s \bm{L}_{j+1/2}^{s} \cdot
\left(\bm{U}_{j-2}-5\bm{U}_{j-1}+10\bm{U}_{j}-10\bm{U}_{j+1}+5\bm{U}_{j+2}-\bm{U}_{j+3} \right). \\
\end{split}
\label{linearupwind}
\end{equation}
It is obvious that this linear scheme omits the computation of the nonlinear weights as presented in Eq.~\eqref{eq8} and ~\eqref{reconstruction},
and decreases by $5/6$ the characteristic-projection operations of Eq.~\eqref{eqchara} as noted in Ref.~\cite{hu2015efficient}.
Therefore, the hybridization will save much computational time.

The threshold is defined as
\begin{equation}
\epsilon=C\left( \frac{\Delta x}{L}\right)^{\alpha} , \\
\label{eq31}
\end{equation}
where $L$ is the characteristic length scale of the problem, $\alpha$ is a positive integer and $C$ is a positive constant.
In this paper, we choose $C=1$ and $\alpha=3$.
When $\sigma^{s}<\epsilon$, we think the flow field is smooth enough and numerical flux is evaluated by the optimal linear upwind scheme.
Otherwise, it is obtained by the regularized WENO-IS scheme.

\section{Numerical cases }
\label{sec1}
In order to assess the performance of the proposed scheme, various test cases are considered,
including shock tube problems, shock/entropy wave interaction problems and a problem involving very strong discontinuities.
In this section, the local Lax-Friendrichs flux is adopted for flux splitting in one dimensional cases,
while Lax-Friendrichs flux is used for two dimensional cases if not mentioned otherwise.
The third order TVD Runge-Kutta scheme is utilized for time integration with a CFL number of 0.5.
In the following, "WENO-JS" denotes the classical 5th-order WENO scheme~\cite{jiang1996efficient},
"WENO-Z" represents the scheme devised in Ref.~\cite{borges2008improved},
"WENO-IS" denotes the original scheme in Ref.~\cite{wang2018incremental},
"WENO-HY" is the hybrid method devised by Hu et al.~\cite{hu2015efficient},
"Present" is the proposed scheme of this paper, and "Exact" denotes the theoretical or convergent solution.

\subsection{Propagation of broadband sound waves }
\label{subsec1}

This case is taken form Ref.~\cite{sun2011class}.
 A propagation of sound wave packet which contains various length scales of acoustic turbulent structures is computed.
The initial condition is given as
 \begin{equation}
 \begin{split}
 &p\left( x,0\right)=p_0\left(1+ \varepsilon  \sum\limits_{k=0}^{N/2}
 \left( E_p\left(k \right) \right)^{1/2} sin\left( 2\pi k \left(x+ \phi_k \right) \right)    \right),     \\
 &\rho\left( x,0\right)=\rho_0\left(p\left(x,0 \right) /p_0 \right)^{1/{\gamma}}    ,\\
 &u\left( x,0\right)=u_0+\frac{2}{\gamma-1}\left(c\left( x,0\right)/c_0  \right)    ,\\
 \end{split}
 \label{wave-propagation}
 \end{equation}
where
 \begin{equation}
E_p\left( k\right)=\left(\frac{k}{k_0} \right)^4 exp^{-2\left( k/k_0\right)^2 }       \\
\label{wave-propagation1}
\end{equation}
is the energy spectrum which reaches its maximum at $k=k_0$. Here, $\phi_k$ is a random number
between $0$ and $1$, $\varepsilon=0.001$, $\gamma=1.4$ and the sound speed $c=\sqrt{\gamma p/\rho}$.
The computational domain is $x\in\left[ 0,1\right] $ and period boundary conditions are imposed at the boundaries.
The computing is executed on a 128-point grid for one period of time with CFL number of $0.2$. In this case, we choose $k_0=12$,
 which means most part of the energy is concentrated on high wavenumbers~\cite{sun2011class}.
 The numerical result of pressure distribution is shown in Fig.~\ref{wave-propagation}.
  We can find that the original WENO-IS scheme produces more excessive numerical dissipation for the high wavenumber waves
  although it has better numerical stability property as stated in Ref.~\cite{wang2018incremental}.
However, with the modification of this paper, the present method achieves better resolution properties
 than WENO-Z and WENO-JS schemes, especially in the region near critical points.
\begin{figure}
	\centering
	{\includegraphics[width=0.6\textwidth]{wave-propagation.png}}
	\caption{ Broadband wave propagation: density profile with $k_0=12$. }
	\label{wave-propagation}
\end{figure}

\subsection{Shock-tube problems }
\label{subsec1}
In this test case, we show the proposed regularized WENO-IS scheme passing the shock tube problems,
which include the Sod problem~\cite{sod1978survey}, the Lax problem~\cite{lax1954weak} and the 1-2-3 problem~\cite{einfeldt1991godunov}.
These computations are executed on a 200-point grid.

The initial condition for Sod problem is
 \begin{equation}
 \left( \rho, u, p\right) =\left\lbrace
 \begin{array}{rcl}
 \left( 1,0,1\right) &{0 \leq x<0.5}\\
 \left( 0.125,0,0.1\right) &{0.5\leq x<1}\\
 \end{array} \right.
 \label{sod}
 \end{equation}
 and the final time is $t=2$.

The initial condition for Lax problem is
\begin{equation}
\left( \rho, u, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 0.445,0.698,3.528\right) &{0 \leq x<0.5}\\
\left( 0.5,0,0.5710\right) &{0.5\leq x<1}\\
\end{array} \right.
\label{Lax}
\end{equation}
and the final time is $t=0.14$.

The initial condition for 123 problem is
\begin{equation}
\left( \rho, u, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 1,-2,0.4\right) &{0 \leq x<0.5}\\
\left( 1,2,0.4\right) &{0.5\leq x<1}\\
\end{array} \right.
\label{123}
\end{equation}
and the final time is $t=1$.

\begin{figure}\centering
	\subfigure {\includegraphics[width=0.45\textwidth]{sod-u.png}}
	\subfigure {\includegraphics[width=0.45\textwidth]{sod-rho.png}}
	\subfigure {\includegraphics[width=0.45\textwidth]{lax-u.png}}
	\subfigure  {\includegraphics[width=0.45\textwidth]{lax-rho.png}}
	\subfigure {\includegraphics[width=0.45\textwidth]{123-u.png}}
	\subfigure  {\includegraphics[width=0.45\textwidth]{123-rho.png}}
	\caption{  Shock tube problems: velocity and density profiles of Sod (top), Lax (middle) and 123 problems (bottom).   }
	\label{1d-shock tube}
\end{figure}

The comparison of the numerical results of density and velocity distribution is shown in Fig.~\ref{1d-shock tube}.
The "Exact" denotes the theoretical solution.
It is obvious that the present method performs as well as other WENO schemes.

\subsection{Interaction blast waves }
\label{subsec1}
A two-blast-wave interaction problem ~\cite{woodward1984numerical} is considered and the initial condition is
\begin{equation}
\left( \rho, u, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 1,0,1000\right) &{0 \leq x<0.1}\\
\left( 1,0,0.01\right) &{0.1\leq x<0.9}\\
\left( 1,0,100\right) &{0.9\leq x<1}.\\
\end{array} \right.
\label{two-blast}
\end{equation}
The Roe flux is adopted for flux splitting and a reflection boundary condition is applied at $x=0$ and $x=1$.
The computing is carried on up to $t=0.038$ on a 400-point grid.
\begin{figure}\centering
	\subfigure {\includegraphics[width=0.45\textwidth]{blast-g.png}}
	\subfigure {\includegraphics[width=0.45\textwidth]{blast-l.png}}
	\caption{  Interacting blast waves: density profile (left) and a zoom of the density profile (right).   }
	\label{blast-wave}
\end{figure}
Fig.~\ref{blast-wave} gives the comparison of density profile and a enlarged part.
The "Exact" denotes the convergent result which is computed by WENO-JS at the resolution of $N=2500$.
It can be observed that all methods capture the strong shock waves very well.
However, the original WENO-IS scheme show considerate larger numerical dissipation and the present method exhibits comparatively less dissipation than others,
especially near the right peak of the density profile as shown in Fig.~\ref{blast-wave}.

\subsection{The Shu-osher problem}
\label{subsec1}
This case gives a Mach 3 shock wave interaction with a sine entropy wave and the initial condition is
\begin{equation}
\left( \rho, u, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 3.857143,2.629369,10.3333\right) &{x\leq 1}\\
\left( 1+0.2sin\left(5x \right) ,0,1\right) &{otherwise}.\\
\end{array} \right.
\label{shu-osher}
\end{equation}
The computing is carried on a 200-point grid with the domain of $x\in\left[0,10 \right] $ and the finial time is $t=1.8$.
The density profiles computed by different schemes are shown in Fig.~\ref{shu-osher}.
The reference "Exact" denotes a high-resolution result solved by WENO-JS on a 3200-point grid.
It can be clearly noticed that all schemes can capture the shock waves well.
However, in the region downstream the shock wave, the present method shows superior resolution
than WENO-Z, WENO-JS and WENO-IS schemes in reproducing the fine flow structures.
Also, compared with TENO5 and TENO5-opt scheme as shown in ~\cite{fu2016family}(their Fig. 15),
the density profile of the present method is more close to the convergent density profile especially in the region downstream the shock wave.
\begin{figure}\centering
	\subfigure {\includegraphics[width=0.45\textwidth]{shu-osher-g.png}}
	\subfigure {\includegraphics[width=0.45\textwidth]{shu-osher-l.png}}
	\caption{  Shu-osher problem: density profile (left) and a zoom of the density profile (right).   }
	\label{shu-osher}
\end{figure}

\subsection{Double Mach reflection of a strong shock }
\label{subsec1}
This is a two-dimensional test case taken from Woodward and Colella~\cite{woodward1984numerical}
on the double Mach reflection of a strong shock.
The initial condition is
\begin{equation}
\left( \rho, u,v, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 1.4,0,0,1.0\right)\quad  &{y\leq 1.732\left(x-0.1667 \right) }\\
\left( 8.0,7.145,-4.125,116.8333\right) &{otherwise},\\
\end{array} \right.
\label{double-mach}
\end{equation}
and the computation is solved on the domain of $\left[ 0,4\right] \times \left[0,1 \right]  $.
 This case represents a right moving shock wave with Mach 10 initially locates at $x=0.1667$, $y=0$
 and makes an angle of $60^{\circ}$
with $x-$axis. For the lower boundary, the reflection wall condition is imposed for $0.1667<x<4$,
while the post shock condition is applied for $0<x<0.1667$.
The computation is carried on two grids with solutions of $512\times 128$ and $1024\times 256$ and the final time is $t=0.2$.

The density profiles on $512\times 128$ and $1024\times 256$ grids
are shown in Fig.~\ref{double-mach512} and Fig.~\ref{double-mach1028}, respectively.
The close-up view of the "blow up" region is presented to compare the fine flow structure resolving abilities of different schemes.
It can be noticed that all methods can capture the main flow features, such as the Mach stem and the near wall jet.
However, the fine structures around the slip line and the near wall jet are considerably different in their solutions.
Among these schemes, the present method resolves the finest flow structures and strongest near wall jet.
Although WENO-Z scheme resolves nearly the same amount of roll-up structures with the present method,
it is inferior in terms of the direction and the shape of the near wall jet compared with
the results with very high resolutions, e.g. Fig.8 in Ref.~\cite{han2011wavelet},
Fig.2.1 in Ref.~\cite{shi2003resolution} and Fig.20 in Ref.~\cite{gerolymos2009very}.
Table~\ref{double-mach-time} gives the CPU time for the entire computing, which counts from the initialization to output,
of double Mach reflection problem  with different methods on the $512\times 128$ grid.
 We can find that the present method is computational efficient
 and only costs less than three fourth of WENO-Z.

\begin{figure}\centering
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-js-512.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-hy-512.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-z-512.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{present-512.png}}
	\caption{  Density profiles of double Mach reflection of a strong shock on the $512\times 128$ grid.
		43 density contours between 1.887 and 20.9 are shown in this figure.   }
	\label{double-mach512}
\end{figure}

\begin{figure}\centering
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-js-1024.png}}
    \subfigure {\includegraphics[width=0.48\textwidth]{weno-hy-1024.png}}
    \subfigure {\includegraphics[width=0.48\textwidth]{weno-z-1024.png}}
    \subfigure {\includegraphics[width=0.48\textwidth]{present-1024.png}}
	\caption{  Density profiles of double Mach reflection of a strong shock on the $1024\times 256$ grid.
		43 density contours between 1.887 and 20.9 are shown in this figure.   }
	\label{double-mach1028}
\end{figure}

\begin{table}
	\scriptsize
	\centering
	\caption{Computational time for double Mach reflection of a strong shock on the $512\times 128$ grid.}
	\begin{tabularx}{13.5cm}{@{\extracolsep{\fill}}llllr}
		\hline
		\quad & WENO-JS &  WENO-HY &  WENO-Z&  Present\\
		\hline
		CPU time (s) & $1721$	& $1146$& $1986$& $1446$ \\
		\hline
	\end{tabularx}
	\label{double-mach-time}
\end{table}

\subsection{Two-dimensional viscous shock-tube problem }
\label{subsec1}
This viscous shock tube problem is taken from Ref.~\cite{daru2000evaluation} and
has been investigated in Refs.~\cite{sjogreen2003grid,zhou2018grid}.
In this problem, the shock wave reflected from the end wall interacts with the boundary layer
on the side wall induced by the incident shock.
These interactions result in a complex system of vortices, shock wave bifurcation and other various flow structures.
The flow filed containing various fine structures makes it a perfect case for testing high-order schemes.
The initial condition is
\begin{equation}
\left( \rho, u,v, p\right) =\left\lbrace
\begin{array}{rcl}
\left( 120,0,0,120/\gamma    \right)  &{0\leq x<1/2 }\\
\left( 1.2,0,0,1.2/\gamma    \right) &{1/2\leq x<1},\\
\end{array} \right.
\label{viscous shock-tube}
\end{equation}
where $\gamma =1.4$ with ideal gas used. The non-slip adiabatic condition is applied to all boundary conditions of the tube.
Since the configuration is symmetric about the line $y=0.5$, only half of the domain is computed,
which is $\left[0,1 \right] \times \left[0,0.5 \right]  $.
Here, we set the Prandtl number $Pr=0.73$ and the final time is $1.0$.
The viscosity is assumed to be constant and the Reynolds number of 200 and 1000 are considered.

The density profiles of the problem with the Reynolds number of 200 are shown in Fig.~\ref{viscous-shock-250}.
The grid converged density profiles in Sj{\"o}green and Yee~\cite{sjogreen2003grid}(their Fig.2) and
Zhou et al.~\cite{zhou2018grid}(their Fig.3) obtained with much higher grid resolutions
are taken as the reference for comparison.
Considering the position of the triple point, the shape and the height of the primary vortex and the orientation of the long axis of the primary vortex,
it is obvious that the present method gives considerable better result than other schemes.
The comparison of the entire computational time of this case by various methods is shown in Table ~\ref{viscous-tube-250}.
We can see that the present method is more efficient than WENO-JS and WENO-Z scheme and only costs nearly three fourth of WENO-Z in this case.

Fig.~\ref{viscous-shock-1028} gives the density profile of the problem with the Reynolds number of 1000.
The grid convergent result can be found in Ref.~\cite{zhou2018grid}(their Fig.6) and Ref.~\cite{fu2016family}(their Fig.21).
With the Reynolds number increased to 1000, more fine flow structures appear in the flow field.
As shown in Fig.~\ref{viscous-shock-1028}, the present method performs well in terms of shock capturing and fine-structure resolution.
We can see that the lambda-shape shocks captured by the present method and the WENO-Z scheme are more sharply than that of WENO-JS and WENO-HY.
The big rotating structures at the lower right corner obtained by WENO-JS, WENO-HY, WENO-Z and TENO5
(see Fig.22 in Ref.~\cite{fu2016family}) are obviously different from the reference solution.
However, the one produced by the present method fits the reference well.
Furthermore, the primary vortex, the adjacent small vortices as well as the jet beneath the oblique shock are predicted quite well with the present method.

\begin{figure}\centering
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-js-250-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-hy-250-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-z-250-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{present-250-vc.png}}
	\caption{ Viscous shock tube problem: 20 density contours between $15$ and $125$. This result is solved on the $250\times125$ grid.  }
	\label{viscous-shock-250}
\end{figure}

\begin{table}
	\scriptsize
	\centering
	\caption{Computational time for viscous shock tube problem on the $250\times 125$ grid.}
	\begin{tabularx}{13.5cm}{@{\extracolsep{\fill}}llllr}
		\hline
		\quad & WENO-JS &  WENO-HY &  WENO-Z&  Present\\
		\hline
		CPU time (s) & $1690$	& $1176$& $1740$& $1343$ \\
		\hline
	\end{tabularx}
	\label{viscous-tube-250}
\end{table}

\begin{figure}\centering
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-js-1028-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-hy-1028-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{weno-z-1028-vc.png}}
	\subfigure {\includegraphics[width=0.48\textwidth]{present-1028-vc.png}}
	\caption{ Viscous shock tube problem: 20 density contours between $20$ and $115$. This result is solved on the $1028\times640$ grid.  }
	\label{viscous-shock-1028}
\end{figure}

\subsection{Mach 2000 jet flows }
\label{subsec1}
 In order to verify the outstanding numerical stability of the present method,
 we consider the Mach 2000 jet problem, which has been computed in Zhang and Shu~\cite{zhang2010positivity} with positive preserving limiters.
The computing is carried out on the domain $\left[0,1 \right]\times \left[ 0,0.25\right]  $ with the resolution of $640\times 160$.
The initial condition is $\left(\rho,u,v,p \right)=\left(0.5,0,0,0.4127 \right)  $ for the entire domain.
A reflective condition is applied at the bottom boundary and an outflow condition is used at the right and top boundaries.
For the left boundary, an inflow condition is adopted with states $\left(\rho,u,v,p \right)=\left(5,800,0,0.4127 \right)  $
when $y<0.05$ and $\left(\rho,u,v,p \right)=\left(0.5,0,0,0.4127 \right)  $ otherwise.
We set $\gamma=5/3$ and the speed of the jet is 800, which gives about Mach 2100 with respect to the sound speed in the jet gas.
In the computing, the CFL number is set to $0.25$ and the final time is $0.001$ as in the reference~\cite{hu2013positivity}.
Note that, without additional positive preserving method applied, the WENO-JS, WENO-HY and WENO-Z schemes
all blow out during the computing.
However, WENO-IS and the present method perform well.
The computed density and pressure profiles of WENO-IS and the present method are shown in Fig.~\ref{ma2000-iso-present}.
We can observe that these results are in good agreement with those in Zhang and Shu~\cite{zhang2010positivity} (their Fig.4.6)
and Hu et al.~\cite{hu2013positivity}(their Fig.4).
Furthermore, the present method shows better resolution than WENO-IS scheme and
the classical WENO scheme combined with positive preserving flux limiters~\cite{zhang2010positivity,hu2013positivity}
with more fine flow structures.
\begin{figure}\centering
	\subfigure {\includegraphics[width=0.85\textwidth]{ma2000-iso-rho.png}}
	\subfigure {\includegraphics[width=0.85\textwidth]{ma2000-rho.png}}
	\subfigure {\includegraphics[width=0.85\textwidth]{ma2000-iso-p.png}}
	\subfigure {\includegraphics[width=0.85\textwidth]{ma2000-p.png}}
	\caption{  Ma 2000  jet problem: 30 density contours of logarithmic scale between $-4$ and $4$(upper);
		30 pressure contours of logarithmic scale between $-1$ and $13$(lower). }
	\label{ma2000-iso-present}
\end{figure}

\section{Conclusions}
\label{sec1}
In this paper, a L2-norm regularized incremental-stencil WENO scheme is proposed for compressible flows.
Following the stencil selection of WENO-IS scheme, we make the following modifications:
(a) a L2-norm error term is introduced in the weighting strategy of WENO-IS scheme
to cope with the discrepancy of the smoothness indicators between 2- and 3-point stencils;
(b) a suitable non-dimensional regularization parameter is chosen to adaptively control the contribution of the error and regularized terms;
(c) a hybridization with the optimal linear scheme is introduced to improve the performance
of the scheme at smooth region and further improve computational efficiency.
These modifications make the method can predict more fine flow structures
by reducing numerical dissipation,
and the remaining of 2-point stencils helps to compute the flow field with strong discontinuities.
A set of test cases reveal that
the proposed method has good fine-structure resolving capability and
keeps the superior robustness.

\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgement}
The first author is partially supported by Xidian University (China) and
the project of Natural Science Foundation of Shaanxi Province (Grant No:2019JM-186).

\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


Biological polypeptides are complex copolymers that derive their phenomenal properties from precisely controlled sequences and compositions of the constituent amino acid monomers, which in turn lead to precisely controlled chain shapes and self-assembled structures. While these materials can be synthetically produced, they are made in notoriously low yield and small quantities.
Polypeptoids are a class of non-natural biomimetic oligomers based on an N-substituted glycine backbone (Fig. 1) that combine many of the advantageous properties of bulk polymers with those of synthetically produced proteins. As opposed to polypeptides, polypeptoids have an achiral backbone devoid of hydrogen bond donors, which makes them readily processable while still being able to form secondary structures such as helices. Furthermore, they are protease resistant, chemically and thermally stable, and have a similar polarity and side chain spacing to that of proteins.
Fig. 1: Polypeptoids move the 'R' group of a peptide from the chiral carbon to the nitrogen.
These polymers can easily be synthesized via a two step submonomer synthesis method that readily incorporates primary amines to obtain side chain functionalities. As a result, literally hundreds of side chain moieties are available for inclusion into the polymer chain. In addition, a rapid, automated synthesizer has recently been developed, making large, scalable batch sizes achievable.
Fig. 2: Polypeptoids may be synthesized with precise sequence control utilizing almost any primary amine.
Our group has created self-assembling polypeptoid systems which have hierarchical levels of control reminiscent of biological molecules. We are currently studying the effect of monomer sequence on the coil-globule transition of peptoid copolymers in solution. In addition, the material properties of polypeptoids are largely unexplored; thus, we are also studying the self-assembly of peptoid-polymer hybrids in the bulk. Polypeptoids are an ideal model system for exploring the effect of monomer sequence on polymer and self-assembly properties.
Biofouling of ship hulls results in highly increased fuel consumption, which is why antifouling coatings are standardly applied on most ships. The first generation antifouling coatings were copper-based, and had a highly negative impact on the marine environment. Because of this, more environmentally friendly alternatives to these coatings have been widely investigated. Present antifouling strategies include the use of highly hydrophilic and even zwitterionic polymers. These polymers inhibit the adsorption of fouling organisms due to the presence of a water hydration layer. On the other hand highly hydrophobic coatings from which foulants can easily be removed by shear forces (so-called fouling release coatings) are also used. Furthermore, three dimensionally patterned surfaces, and even surfaces with mixed functionalities have been proven to discourage attachment. This diversity of strategies results in a large number of vastly different polymer chemistries which are difficult to directly compare, presenting a fundamental problem in gaining understanding and development of coatings.
The high yield and low cost of polypeptoids suggest that they will be ideal components to large scale materials applications including coatings. Many known antifouling functionalities can be incorporated using a single backbone chemistry, making a direct comparison of the effects of different functionalities possible. Moreover, due to the sequence specificity of polypeptoids, different functionalities can be inserted at precisely known locations along the polymer chain, opening up the opportunity to access many sequence specific variants with ease. In this way we can design a coating with optimum performance, while at the same time exploring the chemical and physical properties of these new materials.
Click here to access a list of our publications related to bioinspired polymers.


text:
As Harrisburg exits receivership, a bid to leave Act 47, too
Shown is the Pennsylvania state capitol building ahead of Gov. Tom Corbett's scheduled budget address for the 2014-15 fiscal year to a joint session of the Pennsylvania House and Senate on Tuesday, Feb. 4, 2014, in Harrisburg, Pa. (Matt Rourke/AP Photo, file)
After more than two years under the watchful eye of state officials seeking to deliver it from hundreds of millions of dollars in debt, Pennsylvania's capital city has left state receivership.
A state judge ruled the city has emerged from fiscal emergency as the result of a bond deal to unload its most oppressive debt. Harrisburg remains in the state's Act 47 program for financially distressed municipalities, but Mayor Eric Papenfuse said the state receiver's team has put the city on a path to leave that program, too.
"I think it is absolutely achievable. I certainly hope and expect it to be within my first term," said Papenfuse. "And I think that will be another milestone of almost unprecedented proportions for a third-class city like Harrisburg to emerge."
Harrisburg entered state receivership upon passage of a state law in 2011 empowering the governor to declare a fiscal emergency in the city and put a court-approved monitor in charge of its recovery.
At the time, the law's author, former Republican state Sen. Jeffrey Piccola of Dauphin County, said it gave some teeth to the state's Act 47 program. On Wednesday, Gov. Tom Corbett agreed the law was instrumental in spurring Harrisburg's fiscal recovery.
"I think we've set somewhat of a template of how we can go through this process and hopefully get these cities out of Act 47 as we work on trying to grow their economy and deal with the issues that they're facing," said Corbett. But he added that he doesn't think state receivership is the missing tool for other cities still in Act 47.
Neither does Harrisburg's receiver, William Lynch, a retired Air Force major general. Lynch said he doesn't favor the "concept" of state takeover to bring cities out of Act 47, adding that Harrisburg's debt situation was unique – tied up in multiple refinancings of a trash incinerator, among other things. Other cities in Act 47 face different challenges – namely, underfunded municipal pension obligations.
Pittsburgh Mayor Bill Peduto and state Auditor General Eugene DePasquale are calling on state lawmakers to help municipalities address unfunded pension liabilities. DePasquale said that, taken together, Pennsylvania's municipal unfunded obligations amount to $6.7 billion.
"Pittsburgh, Scranton, Johnstown, New Castle, and others – all of those are [Act} 47 communities, and all of those have significant pension liabilities," said Fred Reddig, who oversees the Act 47 program from within the state's Department of Community and Economic Development.
By contrast, Harrisburg's pension fund is relatively sound, Reddig said.
Six boroughs have emerged from Act 47, but no city of Harrisburg's size has ever left the program after entering it. "It's possible," Reddig said, when asked if the capital city could wipe its hands clean of the program anytime soon. "I'm saying, the next four years – it is possible, yes."

redpajama_set_name: RedPajamaCommonCrawl


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Humans. Over 6000 languages, but I'll wager that only German has the *mot juste* to describe the-performance-of-an-action-with-brute-force-whilst-using-the-utmost-delicacy.
Actually it's not necessary. The Thorens TD 124 is better than any word.
TD 124 Mk 1, Serial#38259. In the centre of the lovely chunky speed selector we see the star-shaped fine speed control wheel. Note the range of voltages and frequencies that can be achieved!
Underneath its beautifully rounded cast frame there is much to describe. What follows is necessarily brief.
It's an idler drive. But not any idler drive. Unlike the legions of cheaper idlers this one was built to be the best. Almost sixty years later many of us still think it's just that.
A top-down approach reveals that there are not one, but two platters. The topmost is a relatively lightweight affair, aluminum with an attached rubber mat and captive 45RPM large hole adapter sprung in place. Underneath this is a simply massive cast platter sporting a superbly machined and polished spindle and bearing (original literature suggests that the machining tolerance for the spindle was 1/1000mm). Why two platters? It's to do with the large motor. When starting the TD 124 from cold the motor will run a percent or so slow for the first couple of minutes and then will run perfectly at speed (possibly for ever given the size of the motor). To avoid speed anomalies (and to achieve a sort of instant cueing for radio stations) the user would keep his or her TD 124 running continuously, not switching it on or off, and raise or lower the top platter to start or stop a record whilst keeping the lower platter spinning at a constant speed. Swiss engineering….
And a little more than half of the top aluminum platter….
Closer view of the ribbed top platter mat, showing the spring-loaded 45RPM large hole adapter, and at the right hand edge the clutch to raise the top platter.
It is an idler drive. Detractors, and there are some, don't approve of the idea of large motors not being isolated from platters. Fair point (kinda), and it is partially the job of the second platter to provide isolation. But this wasn't enough in the view of the engineers at Sainte-Croix. Unlike the wonderful Garrards (301 and 401) and the big Lencos, all beautifully built idlers, the 124 uses a belt between the motor and the idler mechanism to provide even further isolation. It all adds up to a really rather marvelous turntable. Well of course it's still revered! How could it not be.
Underneath the platter(s), showing the spindle well (sintered bronze) at bottom centre. From upper left to right: reversible motor pulley (noting two very small balance holes), belt, stepped idler pulley and brand-new idler wheel. Note the large magnet next to the idler pulley; using the principle of eddy-current braking, fine control of speed is achieved.
A close view of pulleys and eddy-current braking magnet.
Underneath The Beast. Simple, elegant, and extremely strong. Built to last a century?
An examination of the motor pulley reveals tiny drilled holes for balance – and it's reversible for 50Hz or 60Hz use (a lovely touch): when spinning it's actually hard to see that it's moving, such is the accuracy of the machining. The same could be said of the idler wheel itself. Big motor, massive platter, lovely clunky switchgear and nth degree engineering come together to produce the Thorens TD124.
An easily replaced arm board, attached in place with three heavily chromed screws, and the joy is complete.
An exceedingly rare unipivot arm, complete with dedicated Thorens armboard. The signal is conducted through the arm pillar using four small wells of mercury: excellent and highly innovative engineering, but we don't think it will be produced again….Ah, the '60s…..
Of course the original plinth does not date quite so well, being somewhat perfunctory and thin, but there is a veritable army of plinth makers (plinthiers?) all with their own ideas of how best to support and display the TD 124; the one illustrated is a nice stepped ply wood affair, all the way from sunny Moldova, and does its job quite beautifully.
Front view showing the multiple layers of birch ply for the plinth.
A typical original plinth. Not much in the way of real isolation and support; best improved…..
A typical replacement plinth; solid, effective, and so much better than the original flimsy box. The stepped design is designed to safely increase overall density whilst allowing for sufficient air circulation around the motor.
So, a brief peep at one of the Legends. Much, much more to write but in the interests of brevity I'll cut it short for now. Almost 60 years old, and so well put together I suspect that it will last another 60 years. If only I'd been built by the Swiss….
N.B. Keeping the TD 124 Alive For Ever.
Working on the TD 124 and upgrading it is relatively easy. Aside from rebuilding the motor (which requires some precise work) refurbishing the turntable is not terribly difficult. Having rebuilt some basket cases and returned them to their former glory I can say that this is one of the most rewarding undertakings in turntable-world. Using one at home (in fact I'm listening to one as I scribble) makes me once again ask how much vinyl playback has improved over the last half-century. My Quad 57 speakers date from about the same time, and it's difficult to imagine better sound. Methinks we knew a thing or two in 1960….
Thanks for this interesting post and great clear photos. One question you might be able to answer: I am working on the TD124/II which I bought in college in 1973 to bring it back to top condition again after about 15 years in storage.
I cleaned wherever I could reach and lubricated the motor shaft (unable so far to get at the two tiny screws to remove the 50-60 cycle pulley). Removed stepped pulley and idler wheel (which look in good shape and cleaned both with alcohol. Replaced the belt today with correct new one. Motor spins fine and, with new belt, so does the stepped pulley and (apparently) the idler wheel (when platters removed).
However, when I replace the platter and switch it on again (e.g. to 78 rpm), the motor and drive doesn't have the strength to turn the platter.
To me, whatever spring or tensioner holds the idler pulley against the stepped wheel doesn't seem to have the strength needed to hold close enough contact between the two, thus power is being lost in transmission.
Any suggestions? if there is such a tensioner spring, can it be adjusted or replaced? Or am I barking up the wrong tree entirely?
Appreciate any guidance… I'd ship it to you but am on the other side of the world.
There is a sprung assembly which swings the idler wheel into contact with the platter; this is clearly seen when the deck is turned upside down. It's more than likely that the grease on the main shaft has thickened and hardened, and this is probably what is causing the problem. Disassembly is quite logical. although it might appear intimidating at first – wok slowly and all will be well! – cleaning all the old grease off as you go, and re-greasing where necessary. When you have finished and re-assembled the assembly I have no doubt that all will be well. The beauty of the 124 is that it is so simply made, and robust, that it is a joy to work with; this kind of 'repair' will give you all the confidence you need to keep this wonderful deck running for ever!
Great article. I'm currently restoring a MKII and enjoying it immensely. I have just finished the motor overhaul and installed a set of aftermarket machined sintered bronze bearings as original cast phosphor bronze are not available. It's running for a week to break in, with periodic stops to check the coast-down time and put a drop of oil on the bearings as a precaution. I had to polish the rotor spindles as they were scored when the old bushes dried out, but this inevitably removes some material so the tight manufacturing tolerances are no longer there. I'm also a little worried that the machining of the internal bore could have reduced the number and size of pores for the oiling, but we'll have to see how it performs. To be honest, at the running speed of any part of the drive train, 1/1000mm is overkill (and Swiss pride?). Standard replacement bushings have a clearance of 0.04mm, so a smidge either way won't make any difference. You can also infuse slightly different viscosity of oil if necessary, light for tight clearances and increasingly heavy as the clearance increases. I had some problems finding oil as the bearing manufacturers used additive free SAE 30 oil. I eventually found what I was after as a classic motor oil for vintage engines, totally additive free SAE 30, but there are a lot of different opinions. As the metal is porous and the oil migrates in and out of the pores, additive free is best in my opinion, but I saw one well known Hifi magazine recommending Redline 5W40 motor oil, which has additives for modern engines.
Anyhow, I thoroughly enjoyed your article. Please feel free to e-mail me any additional TD124 information, specifications, etc….


text: Q: How to print the hexadecimal values of a given dmp file content I have some memory dumps (.DMP and .raw windows memory files). I want to print the hexadecimal values of the memory dump's content by reading the memory dump. Because i want to search through a particular hexadecimal header and footer and grab the content. I am suppose to grab the content with using regular expressions and i have no idea about how to get the hexadecimal value by reading the memory dump. I am doing some experiment in file carving. SO please give you creative ideas and codes that i can use to implement this task.

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text:
Scott Pruitt Has Laid Bare the Growing Environmental Schism Within Christianity
The EPA administrator seems to confuse "dominion" with "domination" of the earth.
Emily Atkin
Gage Skidmore/Flickr
This story was originally published by The New Republic and appears here as part of the Climate Desk collaboration.
Pope Francis sparked a media frenzy in 2015 with the release of his second encyclical, Laudato Si. One of the highest forms of official Catholic Church teaching, the document presented the moral case for tackling climate change on behalf of the poor and vulnerable. But there was another, less-noticed purpose for Francis's document: To reject an insidious notion among Christians that God wants humans to exploit the Earth's natural resources.
This notion stems from the book of Genesis, in which God gives man "dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps on the earth." Some Christians wrongly believe that this passage "invites us to subjugate the earth, the savage exploitation of nature would be encouraged, presenting the image of human beings as ruler and destroyer," Francis wrote. "This is not the correct interpretation of the Bible as intended by the Church."
In Genesis, God gives man "dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creeps on the earth."
The idea that "dominion" is a free pass to exploit is nonetheless central to Environmental Protection Agency Administrator Scott Pruitt, who has recently become the de facto spokesperson for the anti-Pope brand of Christian environmentalism. Speaking on the pro-Trump Christian Broadcasting Network last week, Pruitt said his faith is partially what motivates him to remove regulations on higher-polluting sources of energy like coal, oil, and natural gas. "The biblical worldview with respect to these issues is that we have a responsibility to manage and cultivate, harvest the natural resources that we've been blessed with to truly bless our fellow mankind," he said. Those comments attracted a lot of media attention, but it's not the first time he has invoked his faith to defend his policies. "God has blessed us with natural resources," he told Politico in July. "Let's use them to feed the world. Let's use them to power the world. Let's use them to protect the world."
Pruitt's rejection of Pope Francis isn't exactly scandalous—he's a Southern Baptist, not a Catholic. Nick Garland, a senior pastor at the church Pruitt attended in Oklahoma, told E&E News that his congregation is in favor of using natural resources. "In Genesis, he [God] put man in it to have dominion; he says [man] should work over it," he said. "Those are the mandates that he set not just for Adam but mankind." The Southern Baptist Convention also published a resolution in 2007 casting doubt on the science of global warming and declaring that Christians should exercise dominion over the Earth.
But even within Pruitt's denomination, this interpretation of "dominion" has been controversial. The year after the 2007 resolution, several prominent leaders of the Southern Baptist Convention rejected it, saying humans "must be proactive and take responsibility" for climate change. One of the most prominent, almost pope-like Southern Baptist leaders, the recently deceased Billy Graham, also wrote that men should use the Earth's resources "wisely, instead of poisoning or destroying it."
We don't worship the earth; instead, we realize that God gave it to us, and we are accountable to Him for how we use it. After creating Adam, the first man, the Bible says, "The Lord God took the man and put him in the Garden of Eden to work it and take care of it" (Genesis 2:15). God didn't tell him to exploit the world or treat it recklessly, but to watch over it and use it wisely. Like a good ruler, we should seek the welfare of everything God entrusts to us—including the creation. The Bible says, "A righteous man cares for the needs of his animal" (Proverbs 12:10).
The interpretation of "dominion" is not up for debate in most parts of the world, Reverend Dave Bookless, the director of theology at the Christian conservation group A Rocha International, said. "Really, the majority view amongst academic theologians, and most bible scholars, is in line with what the Pope has said," Bookless added. "I would struggle to find, certainly in Europe and most parts of the world, a recognized theologian who would take the kind of view Pruitt is taking now."
The debate over whether God wants humans to exploit the Earth's resources really only came into prominence slightly before the industrial revolution, when humans began spewing more pollution into the environment, Bookless said. "The growth of this view into prominence happens simultaneously with the overexploitation of the Earth's resources," he said. "There was this convenient theology to justify it that said, God has given us dominion, and dominion equals domination, therefore the Earth is for us, and we can do what we like with it."
"There was this convenient theology to justify it that said, God has given us dominion, and dominion equals domination, therefore the Earth is for us, and we can do what we like with it."
Pruitt's interpretation of "dominion" is certainly convenient, given his close ties to industry players and the regulations he has loosened at their request. Rules to clean up coal plants and coal waste, fuel-efficiency rules for cars and trucks, rules to prevent methane leaks from oil and gas—all are on their way out, at Pruitt's doing. Is that because he has held, according to The New York Times, "back-to-back meetings, briefing sessions and speaking engagements almost daily with top corporate executives and lobbyists from all the major economic sectors that he regulates," or is it because of his evangelicalism?
If it is because of his faith, it's a faith that's on the fringes of theology, said Jame Schaefer, a professor of systematic theology at Marquette University. "Pruitt's interpretation [of 'dominion'] is not 'theological' per se—not a reflection based on biblical scholars' findings that result from using exegetical methods of interpretation to try to come as close as possible to the meaning intended when written," she wrote in an email. "Unfortunately, he seems to miss the major point: 'The earth is the Lord's as the Psalmists rejoice!' Earth is not Pruitt's, any leader's, or any nation's for use that degrades, destroys, uses up, or precludes use by others for their necessities of life."
While Pruitt's conflation of "dominion" and "domination" may not be theologically correct, Bookless said, his interpretation is certainly politically savvy. "It will appeal to people who have lost their jobs in industry and want them back," he said. "The narrative will play to Trump's supporters, but it's not a view that's mainstream Christian theology at all."
It's Been One Year of Amazing Scott Pruitt Accomplishments, All of Them Horrible
Amy Thomson and Rebecca Leber
Making America Toxic Again
Rebecca Leber


Jared P. T. Standley, 21, was charged with killing his parents in Buckley last week.
The young man accused of killing both his parents in Buckley last week has been officially charged with two counts of aggravated murder in the first degree.
"This is the first double parricide in Pierce County I can recall," said Prosecutor Mark Lindquist in a press release.
According to the Pierce County Prosecutor's Office, the Pierce County Sheriff's department was called to a home on 162nd St. E. in Buckley for a welfare check.
When deputies arrived they observed a bullet hole through the stained glass on the front door.
Inside the front door, deputies found Steven Standley, dead with a gunshot wound to the head. As deputies entered the home further they observed a body wrapped in a tarp, secured with duct tape. The body was later identified as the defendant's mother, Theresa Standley. She was beaten to death.
Sean Standley, who called for the welfare check, reported his brother, Jared Standley, had been kicked out of the home about a week earlier. Sean received a phone call from his father, Steven, earlier in the day. Steven told him Jared had returned to the home unexpectedly. Steven was concerned and told Sean to call 911 if he did not hear back from Steven in 15 minutes. Sean never heard back from Steven so he called 911.
Law officials were able to locate Jared Standley's rental car at the Comfort Inn in Tumwater. Jared Standley attempted to drive away but he was shot in the shoulder and taken into custody. Jared Standley told detectives he went to the house to murder his parents because he "hates them that much." He described killing his mother, Theresa Standley, first. He said he "bludgeoned her with a bat," striking her in the head several times.
Jared Standley said he attempted to clean up the scene while he waited for his father.
His father arrived home later. He said he was trying to get his father into the house so he could shoot him quietly or stab him with a knife. His father ran into the house and attempted to push the door closed to prevent Jared Standley from entering. Jared Standley shot his father in the head, through the glass on the door.
Standley was ordered held without bail.


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Welcome to the last frontier for Silver Salmon on the fly. From early August to mid-October Aleutian Rivers Angling's Sapsuk River (some call it the Hoodoo River) is home to one of Alaska's healthiest runs of Silver salmon. These are powerful chrome fish up to 18 pounds that aggressively take a fly, test your tackle, and keep you coming back for more.
Our river's long runs and deep pools are perfectly sized for effective flyfishing. You will find a good variety of water suitable for both spey rod and single hander, with experienced professional guides to show you the way. The fish are big, grabby, and fresh from the sea. Camp is perfectly situated in the heart of the best water the Sapsuk has to offer, with great fishing within walking distance.
Our guests average around 10-25 Silver salmon per day. This Silver run is prolific and large for its size with the average Silver being 8-10lbs, and topping out at 15-18lbs.
These Silvers are aggressive and top water fishing can be incredible.
The Sapsuk also has a very late Chum salmon run. In the first weeks of the Silver Salmon run (up to about Aug 23th) you will also have great Chum salmon fishing with sea lice Chums coming in with the Silvers.
We also have some good Dolly Varden fishing this time of year and a few resident Rainbows. You will definitely want to bring your 5 weight to fish for them when you have had your fill of Silvers.


February 3, 2017 / Modified aug 2, 2017 3:23 p.m.
Tucson's demonstrators echoed the fears and hopes of the estimated 3 million people who held more than 600 Women's Marches around the world.
An estimated 15,000 people showed up for the Women's March in Tucson, which took place downtown the day after President Trump's inauguration. The day was cold and windy.
It's a demonstration, so many of the marchers carry signs. The messages express concern over President Trump's proposed agenda on immigration, the environment, health care, women's reproductive rights and more.
After the march ends, everyone takes their signs and goes home. Now what? How does a march turn into a movement?
Although the event was called the Women's March, many men and children participated and brought signs voicing their concerns about the new administration's proposed policies.
Organizers who identify as progressive, liberal and Democrat have an unexpected role model these days.
"What would the Tea Party do?" Kristen Randall says that's her new tag line. But she identifies as a progressive. "Whether or not you agree with [the Tea Party's] politics, it doesn't matter. Their strategy was great and it works," she says.
Indivisible is a national movement that was started by Congressional staffers who witnessed firsthand the effectiveness of the Tea Party's strategy during the Obama years. The Indivisible Guide basically copies that strategy.
Kristen Randall is the organizer of Indivisible Southern Arizona, a grassroots activist group that uses the Tea Party Movement's strategy of showing up and saying no.
"Indivisible is focused on resisting the Trump agenda in a local and defensive way," explains Randall. "You want to pass this? I'm going to say no to that." Indivisible doesn't offer alternative policies. It just tries to block what's being proposed. Since December, 4,300 Indivisible groups have sprouted up across the country.
You want to pass this? I'm going to say no to that.
Randall has a busy job as a hydrologist and she's the mother of two, so spare time is in short supply. But she made time to launch Indivisible Southern Arizona. "I think the reason why I have time to do this is that if I didn't, I would be crying," she says.
Randall's group has close to 1,000 members. Indivisible Southern Arizona's first action took place at Congresswoman Martha McSally's Tucson office. The demonstrators pressured her to hold a Town Hall Meeting. So far, it hasn't happened.
Energy and enthusiasm are high right now, but will the energy still be here six months from now, or four years from now?
Will the energy still be here six months from now, or four years from now?
The nonpartisan website AZ Public Square launched in November to promote citizen engagement in Arizona politics.
Tamara Rala Kreiswirth and Ruth Bunny Davis co-founded a new organization to help Arizonans engage with local politics. It's www.azpublicsquare.org. Last year, Kreiswirth started to feel frustrated hearing people say that their vote didn't matter. They didn't make a connection between bills that were being voted on in the Arizona state legislature and issues going on in their own neighborhoods. So visitors to the web site can read about the bills before they are voted on. Davis, who is a retired lawyer, translates the legal jargon into plain language.
"And they can find their legislator. We provide links," adds Davis.
The site is organized by key issue areas in Arizona: immigration, Native peoples, the environment, education, health and voting rights. The site also includes basic civics information.
"This is what a school board member does. Would you like to become a school board member? Here's how you would do that. All that information is on the site," says Kreiswirth.
Kelly Fryer, CEO of the YWCA of Southern Arizona, addressed the crowd from the stage, exhorting participants to see the march as the beginning of a process to fix what's broken in Arizona.
The YWCA is offering its institutional support to grassroots organizers through their Stand Together Arizona Training and Advocacy Center.
" STAT for short," says Fryer, "as in 'This is work that can't wait.' It's one of the things we've been hearing from these grassroots activists. 'We need to know what else is happening in the state. We need to know what everyone's doing and we need to know what's next.' In February we'll be pulling together the activists who planned the 11 Women's Marches across the state" to come up with a common agenda.


Photographers love Iceland. Even the most disciplined find it irresistible and inevitably run out of film long before the end of their trip. Perhaps the reason is the ever-changing light and landscape. Iceland covers a land area of approx. 100,000 square kilometres, ranging from ice fields and black volcanic deserts to lush green meadows and salt marshes.
As an island far out in the north Atlantic the weather can be both extreme and constantly changing, giving rise to swirling clouds, dazzling shafts of light and bright blue skies. Lakes and ponds are everywhere, and on calm days mirror the landscape. In the winter, ice and snow catch the available light giving a luminous quality to photographs.


text:
Five friends, all affected in some way by a mercury poisoning incident are partying when a freak accident put them on a collision course with an entity that is in no mood to forgive their misgivings.
Genre: Thriller, Horror
Studio: Stone Bench Creations, Pen Studios
Keyword: silent film
Cast: Prabhu Deva, Sananth, Indhuja Ravichandran, Remya Nambeesan, Deepak Paramesh, Shashank Purushotham
An aging doorman, after being fired from his prestigious job at a luxurious Hotel is forced to face the scorn of his friends, neighbours and society.
Dr. Mabuse, the Gambler
Dr. Mabuse and his organization of criminals are in the process of completing their latest scheme, a theft of information that will allow Mabuse to...
A doctor's research into the roots of evil turns him into a hideous depraved fiend.
Intolerance: Love's Struggle Throughout the Ages
The story of a poor young woman, separated by prejudice from her husband and baby, is interwoven with tales of intolerance from throughout history.
The Big Parade
The story of an idle rich boy who joins the US Army's Rainbow Division and is sent to France to fight in World War I, becomes friends with two...
The Crowd
John, an ambitious but undisciplined New York City office worker, meets and marries Mary. They start a family, struggle to cope with marital stress,...
The Tramp struggles to live in modern industrial society with the help of a young homeless woman.
A Trip to the Moon
Professor Barbenfouillis and five of his colleagues from the Academy of Astronomy travel to the Moon aboard a rocket propelled by a giant cannon....
The Passion of Joan of Arc
A classic of the silent age, this film tells the story of the doomed but ultimately canonized 15th-century teenage warrior. On trial for claiming...
The love story of an abused English girl and a Chinese Buddhist in a time when London was a brutal and harsh place to live.
A tramp cares for a boy after he's abandoned as a newborn by his mother. Later the mother has a change of heart and aches to be reunited with her son.
The Golem
This mostly lost film (please check your attic) is often confused with director Paul Wegener third and readily available interpretation of the...
God and Satan war over earth; to settle things, they wager on the soul of Faust, a learned and prayerful alchemist.
Francis, a young man, recalls in his memory the horrible experiences he and his fiancée Jane recently went through. Francis and his friend...
In a futuristic city sharply divided between the rich and the poor, the son of the city's mastermind meets a prophet who predicts the coming of a...
Joyless Street
In 1921, we follow two women - Marie and Grete - from the same poor Viennese neighborhood, as they try to better the lives of themselves and their...
The Aerial
In a futuristic city, the residents have lost their voices to both the omnipotent Mr. TV and the nameless corporation that runs the metropolis.
Silent Movie
Aspiring filmmakers Mel Funn, Marty Eggs and Dom Bell go to a financially troubled studio with an idea for a silent movie. In an effort to make the...
The Triplets of Belleville
When her grandson is kidnapped during the Tour de France, Madame Souza and her beloved pooch Bruno team up with the Belleville Sisters—an aged...
Berlin: Symphony of a Great City
A train speeds through the country on its way to Berlin, then gradually slows down as it pulls into the station. It is very early in the morning,...


Commend: Quality of GB I Beam Standard Hot Rolled I Beam is very well, and sales are very patient. Customer Services: Thank you for supporting China Coal! And hoping to be your long-term partner!
English version of GB/T 3077-1999 Alloy Structure Steels: This standard defines size, shape, weight and allowable tolerance, technical requirement, test method, check and acceptance regulations, packing, mark and quality certificate etc. of hot-rolled and forged alloy structure steels.


Finding out that someone you know has cancer can be extremely difficult. All of a sudden you might be having queries related to cancer and about your conduct should be around them. It can be frightening and stressful in case you're very close to the person. At Fortis Hospitals, you'll be offered with premium care as it's known to be one of the best bone cancer hospitals in India.
Guts and Blood: Cancer is often considered an invisible illness during the initial stage. You can't just see cancer cells, so you might assume it's a fairly clean experience in terms of gore. Well, it's not. Cancer might be invisible from the outside but the treatment and surgeries are not. Surgeries come with their own set of gruesome side effects. Dried blood, stitches, iodine stains, bruises, etc. And let's not forget Chemotherapy- it's not a walk in the park either. All these mean your loved one gets jabbed over and over into the same bump located just under their collar bone. Chemo causes nausea, so vomiting is expected. Followed by drugs may make one's face swell and cause a couple of side effects.
Spotlight shifts: Someone you know so well might be dying, which means the spotlight will be on that person. So please, don't let it take you by surprise if all of a sudden the attention you are used to, stops. It might be a dangerous coping habit but try to weigh the situation practically.
Tension so thick that it can choke you: This is going to be absolute hell. Although everyone is going to deal with it differently. Some might withdraw from reacting and some might just react. You're expected to be always on your toes. You're trying to remain strong for the person you care about.
Some people choose the way to be hopeful and reach every possibility to maintain that ray of hope. It's quite possible to assume that if someone who is optimistic must be living in denial, but that's not the case. Making the most of every day is not everyone's cup of tea. As long as they are receiving medical care, they are probably not in denial and their way of coping with cancer should entirely be respected.
Or a simple "How are you doing today?" also works.
If they look good, let them know. It's very important to raise their self-confidence. It's usually advised to never share stories about family members or friends who have faced a similar fate. If someone tells you they have cancer, you should never tell it to anyone else unless you've been given the permission to.
By equipping yourself with the knowledge of how to talk to the person with cancer, you can be the most helpful to them.
To know more visit us!


Toast the red, white and blue with fireworks, family and fun this Fourth!
Tuesday, March 29, 2016 — Let freedom ring! One of Atlanta's favorite summer traditions, Lenox Square's Legendary Fourth of July, is set to return with a star-spangled bang! For over half a century, Lenox Square has celebrated Independence Day in grand tradition with patriotic activities for the entire family. At over 10,000 fireworks, the show has grown in scope and scale to become one of the largest fireworks displays in the Southeast. This year's activities will include a 15-minute fireworks show, musical entertainment and food concessions. Everyone is invited to experience the city's time-honored tradition and celebrate as Lenox Square lights up the sky on the Fourth of July!
Fireworks scheduled to begin at approximately 9:40 p.m.
WHERE: Lenox Square, 3393 Peachtree Rd. N.E.
*Select restaurants will remain open after 6 p.m. for guests to enjoy during the event and will be announced closer to the event date.