A class of high-order weighted compact central schemes for solving hyperbolic conservation laws

TL;DR

This paper introduces a new class of high-order weighted compact central schemes for hyperbolic conservation laws, combining high accuracy, robustness, and efficiency with a novel limiter to handle discontinuities.

Contribution

The paper develops a high-order weighted compact central scheme that achieves super-compact stencils and bypasses Riemann problems, with a tailored WENO limiter for discontinuities.

Findings

Achieves arbitrarily high order accuracy on a super-compact stencil.

Effectively captures discontinuities without spurious oscillations.

Demonstrates robustness and efficiency through numerical tests.

Abstract

We propose a class of weighted compact central (WCC) schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax-Friedrichs (LxF) scheme and the central conservation element and solution element (CESE) scheme. On every cell, the solution is approximated by a Pth order polynomial of which all the DOFs are stored and updated separately. The cell average is updated by a classical finite volume scheme which is constructed based on space-time staggered meshes such that the fluxes are continuous across the interfaces of the adjacent control volumes and, therefore, the local Riemann problem is bypassed. The kth order spatial derivatives are updated by a central difference of (k-1)th order spatial derivatives at cell vertices. All the space-time information is calculated by the Cauchy-Kovalewski procedure. By doing so, the schemes are able to achieve arbitrarily uniform spacetime high order on a super-compact stencil with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory (WENO) type limiter is tailor-made for the schemes. The limiter preserves the compactness and high order accuracy of the schemes. The accuracy, robustness, and efficiency of the schemes are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.