Amplitude-Shape Method for the Numerical Solution of Ordinary Differential Equations

Abstract

The numerical solution of large stiff systems of ordinary differential
equations is very expensive and often impossible to be done on a PC. Even if
a large mainframe computer is used, it might be too slow to follow the evolution
of a physical system in real time and parallel computations are needed. In
this paper we propose an amplitude-shape method which takes into account the
particular structuure of some evolution problems, especially those described
by partial differental equations. This method consists in transforming the
system so that only a few equations remain stiff, the majority of the equations
are non-stiff. The system is treated with a mixed explicit-implicit scheme
which leads to a considerable reduction of numerical effort. We compare our
approach with a classical solver of ordinary differential equations taking
as an example, stiff systems of equations describing spatially dependent chemical
kinetics.

Keywords: amplitude-shape/ode/numerical/stiff equations/differential
equations/computer
Mathematics Subject Classification (1991): 34A65

Quaestiones Mathematicae
24(1) 2001, 65–73