Improved Sixth-Order WENO Finite Difference Schemes for Hyperbolic Conservation Laws

Abstract

This article describes developing and improving sixth-order characteristic-wise Weighted Essentially Non-Oscillatory (WENO) finite difference schemes. These schemes are specially designed to solve scalar and system hyperbolic conservation laws with high accuracy/resolution and robustness. The schemes have been enhanced by using a new reference global smoothness indicator, which ensures the optimal order of accuracy for smooth solutions. The schemes also incorporate affine-invariant nonlinear Ai-weights that are independent of the scaling of solution and the choice of sensitivity parameter. The improved nonlinear weights enhance the essentially non-oscillatory (ENO) capturing of discontinuities and minimize the numerical dissipation, especially for long-time simulations. The study also introduces the positivity-preserving limiter to ensure that the numerical solution of Euler equations is physically valid. The effectiveness of improved schemes is demonstrated through one- and two-dimensional benchmark shock-tube problems, such as the Sod, Lax, and Woodward-Colella problems. The improved schemes are compared with other WENO schemes in terms of accuracy, resolution, ENO, and robustness.