On Asymptotic Global Error Estimation and Control of Finite Difference Solutions for Semilinear Parabolic Equations

TL;DR

This paper develops a method for estimating and controlling the overall error in finite difference solutions of semilinear parabolic PDEs by combining global error estimation with Richardson extrapolation and adaptive mesh refinement.

Contribution

It extends previous global error control techniques to finite difference solutions of semilinear parabolic equations, incorporating spatial truncation error estimation and asymptotic error transport equations.

Findings

Reliable error estimation demonstrated through numerical examples.

Effective global error control achieved with adaptive mesh refinement.

Asymptotic estimates neglecting higher order errors are valid for small step sizes.

Abstract

The aim of this paper is to extend the global error estimation and control addressed in Lang and Verwer [SIAM J. Sci. Comput. 29, 2007] for initial value problems to finite difference solutions of semilinear parabolic partial differential equations. The approach presented there is combined with an estimation of the PDE spatial truncation error by Richardson extrapolation to estimate the overall error in the computed solution. Approximations of the error transport equations for spatial and temporal global errors are derived by using asymptotic estimates that neglect higher order error terms for sufficiently small step sizes in space and time. Asymptotic control in a discrete $L_2$-norm is achieved through tolerance proportionality and uniform or adaptive mesh refinement. Numerical examples are used to illustrate the reliability of the estimation and control strategies.