Fast weak-KAM integrators for separable Hamiltonian systems

TL;DR

This paper introduces a fast, convergent numerical scheme for Hamilton-Jacobi equations that preserves geometric properties and efficiently computes solutions using min-plus convolutions, suitable for long-term simulations.

Contribution

It develops a new fast integrator for weak-KAM solutions of separable Hamiltonian systems with proven convergence and geometric properties, enhancing computational efficiency.

Findings

The scheme converges with Lipschitz solutions and provides error estimates.

It acts as a geometric integrator satisfying a discrete weak-KAM theorem.

The method enables efficient long-time computations of Hamilton-Jacobi solutions.

Abstract

We consider a numerical scheme for Hamilton-Jacobi equations based on a direct discretization of the Lax-Oleinik semi-group. We prove that this method is convergent with respect to the time and space stepsizes provided the solution is Lipschitz, and give an error estimate. Moreover, we prove that the numerical scheme is a geometric integrator satisfying a discrete weak-KAM theorem which allows to control its long time behavior. Taking advantage of a fast algorithm for computing min-plus convolutions based on the decomposition of the function into concave and convex parts, we show that the numerical scheme can be implemented in a very efficient way.