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

Espen Robstad Jakobsen, Kenneth Hvistendahl Karlsen, and Nils Henrik Risebro

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.

[ 1996 | 1997 | 1998 | 1999 | 2000 | All Preprints | Preprint Server Homepage ]
© The copyright for the following documents lies with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use.

Conservation Laws Preprint Server <conservation@math.ntnu.no>
Last modified: Mon Feb 7 09:21:26 2000