A Kernel Based High Order "Explicit" Unconditionally Stable Scheme for Time Dependent Hamilton-Jacobi Equations

TL;DR

This paper introduces a high order, unconditionally stable kernel-based numerical scheme for Hamilton-Jacobi equations, enabling larger time steps and accurate solutions for complex geometries and boundary conditions.

Contribution

It extends previous kernel-based methods to Hamilton-Jacobi equations, achieving high order accuracy and unconditional stability with enhanced boundary handling.

Findings

Achieves high order accuracy in space and time.

Unconditionally stable allowing larger time steps.

Effective in complex geometries and boundary conditions.

Abstract

In this paper, a class of high order numerical schemes is proposed for solving Hamilton-Jacobi (H-J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. \emph{arXiv preprint arXiv:1707.09294},, which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H-J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax-Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H-J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H-J equations is verified.