Preprint 2000-007

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

Abstract: We establish a rate of convergence for a semi-discrete 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 \cite{maxSou}. 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.

Paper:

Available as PostScript.

Author(s):

Espen Robstad Jakobsen, <erj@math.ntnu.no>

Kenneth Hvistendahl Karlsen, <kennethk@mi.uib.no>

Nils Henrik Risebro, <nilshr@math.uio.no>

Publishing information:

Submitted to Siam. J. Numer. Anal.

Comments:

Submitted by:

<kennethk@mi.uib.no> February 5 2000.

Revised version submitted by:

<erj@math.ntnu.no> August 20 2000.