# Optimal error estimates for semidiscrete phase relaxation models

RAIRO Model. Math. Anal. Numer., 31 (1997), 91-120.

Ricardo H. Nochetto
Department of Mathematics
University of Maryland, College Park
College Park, MD 20742, USA

rhn@math.umd.edu

Xun Jiang
Department of Mathematics
University of California at Davis
Davis, CA 95616, USA

xun@galois.ucdavis.edu

## Abstract

This paper examines and compares semi-implicit, semi-explicit and extrapolation time-discretizations of a simple phase relaxation model with small parameter $\ep$. The model consists of a diffusion-advection-reaction PDE for temperature coupled with an ODE with double obstacle $\pm 1$ for phase variable. Sharp interfaces are thereby replaced by thin transition layers of thickness $\O(\sqrt\ep\,)$. As time-step $\tau\da0$, the semi-implicit and extrapolation schemes are shown to converge with optimal orders $\O(\tau)$ for temperature and enthalpy, and $\O(\sqrt\tau\,)$ for heat flux, irrespective of $\ep$, provided $\tau\le{\ep}/2$ for the extrapolation scheme. The second scheme may be viewed as a linearization of the first. For the semi-explicit counterpart, which is also a linearization subject to the stability constraint $\tau\le\ep$, these orders are further multiplied by an extra factor $1/\sqrt\ep$, and are sharp. The results for the semi-implicit scheme are preserved in the singular limit $\ep\da0$, namely the Stefan problem with temperature-dependent convection and reaction.