A Discontinuous Galerkin Semi-Lagrangian Scheme for 1D Hamilton-Jacobi-Bellman Equations

The Hamilton-Jacobi-Bellman (HJB) equation, due to its nonlinearity, in general does not admit a classical solution, also for regular data. For this reason, the numerical approximation of the solution may pose some additional difficulties, compared to other cases. Over the past four decades, the literature has proposed several numerical approaches. The methods proposed include discontinuous Galerkin (DG) due to its properties of being local, flexible, and robust, but the approach remained underused, due to various technical difficulties, in particular its difficulty in selecting the correct viscosity solution of the problem. In this paper, a numerical method is proposed to solve the evolution HJB equation in one dimension. It consists of the combination of a Semi-Lagrangian (SL) scheme, aimed at reconstructing the characteristic directions, and a DG method, aimed at generating an approximate solution as a linear combination of discontinuous and compactly supported basis functions, defined a priori. In order to evaluate the performance of the proposed method, a collection of numerical experiments with regular, simply continuous, and discontinuous data is presented.