  In this study, a new framework of constructing very high order
discontinuity-capturing schemes is proposed for finite volume method. These
schemes, so-called $\mathrm{P}_{n}\mathrm{T}_{m}-\mathrm{BVD}$ (polynomial of
$n$-degree and THINC function of $m$-level reconstruction based on BVD
algorithm), are designed by employing high-order linear-weight polynomials and
THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive
steepness as the reconstruction candidates. The final reconstruction function
in each cell is determined with a multi-stage BVD (Boundary Variation
Diminishing) algorithm so as to effectively control numerical oscillation and
dissipation. We devise the new schemes up to eleventh order in an efficient way
by directly increasing the order of the underlying upwind scheme using
linear-weight polynomials. The analysis of the spectral property and accuracy
tests show that the new reconstruction strategy well preserves the
low-dissipation property of the underlying upwind schemes with high-order
linear-weight polynomials for smooth solution over all wave numbers and
realizes $n+1$ order convergence rate. The performance of new schemes is
examined through widely used benchmark tests, which demonstrate that the
proposed schemes are capable of simultaneously resolving small-scale flow
features with high resolution and capturing discontinuities with low
dissipation. With outperforming results and simplicity in algorithm, the new
reconstruction strategy shows great potential as an alternative numerical
framework for computing nonlinear hyperbolic conservation laws that have
discontinuous and smooth solutions of different scales.

Computational Physics