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


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



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.