Numerical methods for a differential equation arising in geology

Neville J. Ford, Judith M. Ford, John Wheeler

Department of Mathematics, Chester College, Parkgate Road, Chester, CH1 4BJ, UK

njford@chester.ac.uk

http://www.chester.ac.uk/~neville/njf-real.html

Contributed talk

We consider differential equations of the form

$$y'(t) = f(y),\quad y(0)=y_0,\quad f(y) = f_\epsilon(y) + \mathcal{O}(\epsilon),$$

where the function $f(y)$ cannot be computed explicitly, but can be approximated by $f_\epsilon(y)$ for any given $y$. $\epsilon$ may be a known constant, or it may be possible to define an $f_\epsilon$ for any chosen $\epsilon$. An example of such an equation arises in the modelling of grain boundary diffusion creep [1,3]. We are interested in the convergence and stability of numerical methods for solving these equations and, in particular, in the effect of the $\epsilon$ on the accuracy of the numerical solution. We present a detailed analysis of the convergence and stability of the forward Euler method and of a predictor-corrector method based on a forward Euler predictor and a trapezoidal corrector and discuss some practical implications for the solution of such equations [2].

This work is partially supported by NERC Research Grant NER/B/S/2000/000667