On the convergence rate of operator splitting for Hamilton-Jacobi equations with source terms

摘要

We establish a rate of convergence for a semidiscrete operator splitting method applied to Hamilton Jacobi equations with source terms. The method is based on sequentially solving a Hamilton Jacobi equation and an ordinary differential equation. The Hamilton Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the L-infinity error associated with the operator splitting method is bounded by O(Deltat), where Deltat is the splitting (or time) step. This error bound is an improvement over the existing O (root Deltat) bound due to Souganidis [Nonlinear Anal., 9 (1985), pp. 217-257]. In the one-dimensional case, we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous Hamilton Jacobi equations. It is proved that this fully discrete splitting method possesses a linear convergence rate. Moreover, numerical results are presented to illustrate the theoretical convergence results. [References: 50]