TY - JOUR

T1 - A class of high-order weighted compact central schemes for solving hyperbolic conservation laws

AU - Shen, Hua

AU - Al Jahdali, Rasha

AU - Parsani, Matteo

N1 - KAUST Repository Item: Exported on 2022-06-27
Acknowledged KAUST grant number(s): OSR-2019-CCF-3666
Acknowledgements: H. S. would like to acknowledge the financial support of the National Natural Science Foundation of China (Contract No. 11901602). R. J. and M. P. were supported by King Abdullah University of Science and Technology through the award OSR-2019-CCF-3666.

PY - 2022/6/23

Y1 - 2022/6/23

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/668735

UR - https://linkinghub.elsevier.com/retrieve/pii/S0021999122004326

U2 - 10.1016/j.jcp.2022.111370

DO - 10.1016/j.jcp.2022.111370

M3 - Article

SN - 0021-9991

VL - 466

SP - 111370

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -