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

Abstract

We propose a class of weighted compact central schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax–Friedrichs scheme and the central conservation element and solution element 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 the (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 space-time high-order on a compact stencil consisting of only neighboring cells with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory type limiter is tailor-made for the schemes. The limiter preserves the compactness and high-order accuracy of the schemes. The schemes' accuracy, robustness, and efficiency are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.