Arc length based WENO scheme for Hamilton-Jacobi Equations

TL;DR

This paper introduces a new WENO scheme using arc-length based smoothness indicators for Hamilton-Jacobi equations, improving accuracy and resolution over classical methods through novel polynomial derivative measures.

Contribution

The paper proposes a novel arc-length based smoothness indicator for WENO schemes, enhancing accuracy and resolution in solving Hamilton-Jacobi equations.

Findings

Improved numerical accuracy compared to classical WENO schemes.

Enhanced resolution of solutions through arc-length based smoothness indicators.

Extensive numerical tests validate the scheme's performance.

Abstract

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and gives a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.