Filtered schemes for Hamilton-Jacobi equations: a simple construction of convergent accurate difference schemes

TL;DR

This paper introduces a simple, convergent class of filtered finite difference schemes for Hamilton-Jacobi equations that achieve high accuracy and are efficiently solvable, validated through computational experiments.

Contribution

It presents a new, general framework for constructing high-order accurate, explicit filtered schemes for Hamilton-Jacobi equations, ensuring convergence to viscosity solutions.

Findings

Validated high-order accuracy with computational results

Schemes are explicit and compatible with fast sweeping/marching methods

Demonstrated effectiveness on the eikonal equation in 1D and 2D

Abstract

We build a simple and general class of finite difference schemes for first order Hamilton-Jacobi (HJ) Partial Differential Equations. These filtered schemes are convergent to the unique viscosity solution of the equation. The schemes are accurate: we implement second, third and fourth order accurate schemes in one dimension and second order accurate schemes in two dimensions, indicating how to build higher order ones. They are also explicit, which means they can be solved using the fast sweeping method or the fast marching method.The accuracy of the method is validated with computational results for the eikonal equation in one and two dimensions, using filtered schemes made from standard centered differences, higher order upwinding and ENO interpolation.