Second Order Numerical Methods for First Order Hamilton-Jacobi Equations

We present practical numerical methods which produce provably second order approximations for a class of stationary first order Hamilton--Jacobi partial differential equations. Using probabilistic methods, we derive high order asymptotic expansions for a first order method and then use those results to design second order methods. We prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth. Although we limit our attention to second order schemes, in principle the techniques in this paper can be extended to arbitrarily high order methods. Examples illustrate the rate of convergence as well as global sharp resolution of discontinuities. The Hamilton--Jacobi equations we consider correspond to deterministic optimal control problems, and our rate of convergence results are valid for the value functions and for the optimal feedback controls.

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