A Convergence Analysis of the PeacemanRachford Scheme for Semilinear Evolution Equations
Abstract
The PeacemanRachford scheme is a commonly used splitting method for discretizing semilinear evolution equations, where the vector fields are given by the sum of one linear and one nonlinear dissipative operator. Typical examples of such equations are reactiondiffusion systems and the damped wave equation. In this paper we conduct a convergence analysis for the PeacemanRachford scheme in the setting of dissipative evolution equations on Hilbert spaces. We do not assume Lipschitz continuity of the nonlinearity, as previously done in the literature. First or second order convergence is derived, depending on the regularity of the solution, and a shortened proof for $o(1)$convergence is given when only a mild solution exits. The analysis is also extended to the Lie scheme in a Banach space framework. The convergence results are illustrated by numerical experiments for Caginalp's solidification model and the GrayScott pattern formation problem.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.05925
 Bibcode:
 2015arXiv151205925H
 Keywords:

 Mathematics  Numerical Analysis;
 65J08;
 65M12;
 47H06
 EPrint:
 SIAM Journal on Numerical Analysis, 51(4), 2013, 19001910