On the Convergence Rate of Operator Splitting for Hamilton--Jacobi Equations with Source Terms

Authors:
Espen Robstad Jakobsen;Kenneth Hvistendahl Karlsen;Nils Henrik Risebro
Affiliations:
-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
2001

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Abstract

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 $L^{\infty}$ error associated with the operator splitting method is bounded by $\mathcal{O}(\Delta t)$, where $\Delta t$ is the splitting (or time) step. This error bound is an improvement over the existing $\mathcal{O}(\sqrt{\Delta t})$ 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.