New Finite Difference Mapped WENO Schemes with Increasingly High Order of Accuracy

Abstract

In this paper, a new type of finite difference mapped weighted essentially non-oscillatory (MWENO) schemes with unequal-sized stencils, such as the seventh-order and ninth-order versions, is constructed for solving hyperbolic conservation laws. For the purpose of designing increasingly high-order finite difference WENO schemes, the equal-sized stencils are becoming more and more wider. The more we use wider candidate stencils, the bigger the probability of discontinuities lies in all stencils. Therefore, one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some four-point or five-point stencils into several smaller three-point stencils. By the usage of this new methodology in high-order spatial reconstruction procedure, we get different degree polynomials defined on these unequal-sized stencils, and calculate the linear weights, smoothness indicators, and nonlinear weights as specified in Jiang and Shu (J. Comput. Phys. 126: 202228, 1996). Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions, another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights, so as to keep the optimal order of accuracy in smooth regions. These new MWENO schemes can also be applied to compute some extreme examples, such as the double rarefaction wave problem, the Sedov blast wave problem, and the Leblanc problem with a normal CFL number. Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.

Funding

National Natural Science Foundation of China(12071392); Science Challenge Project(TZ2016002)