A New Type of Modified WENO Schemes for Solving Hyperbolic Conservation Laws

Abstract

In this paper we design a new type of the third order and fifth order modified weighted essentially nonoscillatory (MWENO) schemes in the finite difference framework for solving the hyperbolic conservation laws. These schemes adapt between the linear upwind scheme and the WENO scheme automatically by the usage of a new simple switching principle. The methodology to reconstruct numerical fluxes for the MWENO schemes is split into two parts: if all extreme points of the reconstruction polynomial for numerical flux in the big spatial stencil are located outside of the stencil, the numerical flux is approximated directly by the reconstruction polynomial, and the approximation is linear and of high order accuracy; otherwise, the WENO procedure in [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202--228; C.-W. Shu, SIAM Review, 51 (2009), pp. 82--126] is applied to reconstruct the numerical flux. The main advantage of these new MWENO schemes is their robustness and efficiency compared to the classical WENO schemes specified in [G.-S. Jiang and C.-W. Shu, J. Comput. Phys., 126 (1996), pp. 202--228; C.-W. Shu, SIAM Review, 51 (2009), pp. 82--126.]. The MWENO schemes can be applied to compute some extreme test cases such as the Sedov blast wave, the Leblanc and the high Mach number astrophysical jet problems, etc., by using a normal CFL number without any further positivity preserving procedure for the purpose of controlling the concurrence of negative density and pressure. Extensive numerical results are provided to illustrate the good performance of the MWENO schemes.