A numerical study of the performance of alternative weighted ENO methods based on various numerical fluxes for conservation law

In this paper, we systematically investigate the performance of the weighted essential non-oscillatory (WENO) methods based on various numerical fluxes for the nonlinear hyperbolic conservation law. Our objective is enhancing the performance for the conservation laws through picking out the suitable numerical fluxes. We focus our attention entirely on the comparison of eight numerical fluxes with the fifth-order accurate finite difference WENO methods and third-order accurate TVD RungeKutta time discretization for hyperbolic conservation laws. In addition, we give their implementation based on a new form framework of flux which we used in 9 and was proposed by Shu and Osher in 16. The detailed numerical study is mainly implemented for the one dimensional system case, including the discussion of the CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

[1]  Chi-Wang Shu,et al.  Finite Difference WENO Schemes with Lax-Wendroff-Type Time Discretizations , 2002, SIAM J. Sci. Comput..

[2]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[3]  S. Osher,et al.  Upwind difference schemes for hyperbolic systems of conservation laws , 1982 .

[4]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[5]  Yan Jiang,et al.  An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax-Wendroff Time Discretization for Conservation Laws , 2013, SIAM J. Sci. Comput..

[6]  Jianxian Qiu,et al.  A Numerical Comparison of the Lax–Wendroff Discontinuous Galerkin Method Based on Different Numerical Fluxes , 2007, J. Sci. Comput..

[7]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[8]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[9]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[10]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[11]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[12]  S. Osher,et al.  One-sided difference approximations for nonlinear conservation laws , 1981 .

[13]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[14]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[15]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[16]  Chi-Wang Shu,et al.  A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes , 2006, J. Comput. Phys..

[17]  Jianxian Qiu,et al.  Finite Difference Hermite WENO Schemes for Conservation Laws, II: An Alternative Approach , 2015, Journal of Scientific Computing.

[18]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[19]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[20]  Michael A. Leschziner,et al.  Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows , 1997 .