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We analyse two practical aspects that arise in the numerical solution ofHamilton-Jacobi-Bellman (HJB) equations by a particular class of monotoneapproximation schemes known as semi-Lagrangian schemes. These schemes make useof a wide stencil to achieve convergence and result in discretization matricesthat are less sparse and less local than those coming from standard finitedifference schemes. This leads to computational difficulties not encounteredthere. In particular, we consider the overstepping of the domain boundary andanalyse the accuracy and stability of stencil truncation. This truncationimposes a stricter CFL condition for explicit schemes in the vicinity ofboundaries than in the interior, such that implicit schemes become attractive.We then study the use of geometric, algebraic and aggregation-based multigridpreconditioners to solve the resulting discretised systems from implicit timestepping schemes efficiently. Finally, we illustrate the performance of thesetechniques numerically for benchmark test cases from the literature.

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