Increasingly high-order hybrid multi-resolution WENO schemes in multi-dimensions

Abstract

In this paper, a new type of increasingly high-order hybrid multi-resolution weighted essentially non-oscillatory (HMR-WENO) schemes is presented in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. Based on the reconstruction polynomials defined on the one-point, three-point, five-point, seven-point, and nine-point spatial stencils, we reconstruct one zeroth degree reconstruction polynomial, one quadratic reconstruction polynomial, one quartic reconstruction polynomial, one sextic reconstruction polynomial, one octave reconstruction polynomial, and their derivative polynomials together with a new hierarchical bisection method to design a series of new troubled cell indicators which can precisely find all extreme points of associated unequal degree reconstruction polynomials located inside the smallest interval in one dimension. The new troubled cell indicators do not introduce any manual parameters related to different problems. The new hybrid methodology is divided into two parts: if all extreme points of the reconstruction polynomials are nonexistent or outside the smallest interval, the target cell is not a troubled cell and the simple linear upwind schemes are utilized to obtain high-order approximations. Otherwise, the target cell is a troubled cell and the MR-WENO spatial reconstruction procedures with excellent shock-capture ability are adopted. Then a series of HMR-WENO schemes are proposed by using these new troubled cell indicators, which can be easily expanded to arbitrarily high-order accuracies in multi-dimensions. The main benefits of these HMR-WENO schemes are their efficiency, since they could save about 22%-75% CPU time than that of the same order MR-WENO schemes when simulating some benchmark examples in multi-dimensions.