An efficient filtered scheme for some first order Hamilton-Jacobi-Bellman equations

TL;DR

This paper presents a new class of filtered schemes for first order Hamilton-Jacobi-Bellman equations that are high-order, easy to implement, and provably convergent with error estimates, improving stability and accuracy.

Contribution

The work introduces a novel filtered scheme framework that combines high-order accuracy with convergence guarantees for Hamilton-Jacobi-Bellman equations.

Findings

Schemes are not monotone but satisfy an $psilon$-monotone property.

Convergence and error estimates of order .5 in mesh size.

Numerical tests validate the approach and demonstrate stabilization of high-order schemes.

Abstract

We introduce a new class of "filtered" schemes for some first order non-linear Hamilton-Jacobi-Bellman equations. The work follows recent ideas of Froese and Oberman (SIAM J. Numer. Anal., Vol 51, pp.423-444, 2013). The proposed schemes are not monotone but still satisfy some $\epsilon$-monotone property. Convergence results and precise error estimates are given, of the order of $\sqrt{\Delta x}$ where $\Delta x$ is the mesh size. The framework allows to construct finite difference discretizations that are easy to implement, high--order in the domains where the solution is smooth, and provably convergent, together with error estimates. Numerical tests on several examples are given to validate the approach, also showing how the filtered technique can be applied to stabilize an otherwise unstable high--order scheme.