Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations

TL;DR

This paper investigates boundary treatment and multigrid preconditioning techniques to efficiently solve semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations, addressing computational challenges from wide stencils and boundary overstepping.

Contribution

It introduces analysis of boundary truncation effects and proposes multigrid preconditioning methods to improve computational efficiency for implicit semi-Lagrangian schemes.

Findings

Stencil truncation affects accuracy and stability near boundaries.

Implicit schemes with multigrid preconditioning are effective.

Numerical tests demonstrate improved performance on benchmark problems.

Abstract

We analyse two practical aspects that arise in the numerical solution of Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone approximation schemes known as semi-Lagrangian schemes. These schemes make use of a wide stencil to achieve convergence and result in discretization matrices that are less sparse and less local than those coming from standard finite difference schemes. This leads to computational difficulties not encountered there. In particular, we consider the overstepping of the domain boundary and analyse the accuracy and stability of stencil truncation. This truncation imposes a stricter CFL condition for explicit schemes in the vicinity of boundaries than in the interior, such that implicit schemes become attractive. We then study the use of geometric, algebraic and aggregation-based multigrid preconditioners to solve the resulting discretised systems from implicit time stepping schemes efficiently. Finally, we illustrate the performance of these techniques numerically for benchmark test cases from the literature.