An Improved Non-linear Weights for Seventh-Order WENO Scheme

TL;DR

This paper introduces an improved seventh-order WENO scheme with enhanced smoothness indicators for hyperbolic conservation laws, demonstrating high accuracy and stability in both linear and nonlinear test cases.

Contribution

The paper develops a novel non-linear weighting method for seventh-order WENO schemes using $L_{1}$-norm based smoothness indicators, improving accuracy and robustness.

Findings

Achieves seventh-order accuracy in smooth regions.

Demonstrates stability and convergence in linear and nonlinear tests.

Effective in one- and two-dimensional problems.

Abstract

In this article, the construction and implementation of a seventh order weighted essentially non-oscillatory scheme is reported for hyperbolic conservation laws. Local smoothness indicators are constructed based on $L_{1}$-norm, where a higher order interpolation polynomial is used with each derivative being approximated to the fourth order of accuracy with respect to the evaluation point. The global smoothness indicator so constructed ensures the scheme achieves the desired order of accuracy. The scheme is reviewed in the presence of critical points and verified the numerical accuracy, convergence with the help of linear scalar test cases. Further, the scheme is implemented to non-linear scalar and system of equations in one and two dimensions. As the formulation is based on method of lines, to move forward in time linear strong-stability-preserving Runge-Kutta scheme (lSSPRK) for the linear problems and the fourth order nonlinear version of five stage strong stability preserving Runge-Kutta scheme (SSPRK(5,4)) for nonlinear problems is used.