  We present a new line-based discontinuous Galerkin (DG) discretization scheme
for first- and second-order systems of partial differential equations. The
scheme is based on fully unstructured meshes of quadrilateral or hexahedral
elements, and it is closely related to the standard nodal DG scheme as well as
several of its variants such as the collocation-based DG spectral element
method (DGSEM) or the spectral difference (SD) method. However, our motivation
is to maximize the sparsity of the Jacobian matrices, since this directly
translates into higher performance in particular for implicit solvers, while
maintaining many of the good properties of the DG scheme. To achieve this, our
scheme is based on applying one-dimensional DG solvers along each coordinate
direction in a reference element. This reduces the number of connectivities
drastically, since the scheme only connects each node to a line of nodes along
each direction, as opposed to the standard DG method which connects all nodes
inside the element and many nodes in the neighboring ones. The resulting scheme
is similar to a collocation scheme, but it uses fully consistent integration
along each 1-D coordinate direction which results in different properties for
nonlinear problems and curved elements. Also, the scheme uses solution points
along each element face, which further reduces the number of connections with
the neighboring elements. Second-order terms are handled by an LDG-type
approach, with an upwind/downwind flux function based on a switch function at
each element face. We demonstrate the accuracy of the method and compare it to
the standard nodal DG method for problems including Poisson's equation, Euler's
equations of gas dynamics, and both the steady-state and the transient
compressible Navier-Stokes equations.

Numerical Analysis