Arc Length-Based WENO Scheme for Hamilton–Jacobi Equations

Abstract

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of the 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 give 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.