Stepsize Selection in Explicit Runge-Kutta Methods for Moderately Stiff Problems

Justin Steven Calder Prentice

doi:10.4236/am.2011.26094

Abstract

We present an algorithm for determining the stepsize in an explicit Runge-Kutta method that is suitable when solving moderately stiff differential equations. The algorithm has a geometric character, and is based on a pair of semicircles that enclose the boundary of the stability region in the left half of the complex plane. The algorithm includes an error control device. We describe a vectorized form of the algorithm, and present a corresponding MATLAB code. Numerical examples for Runge-Kutta methods of third and fourth order demonstrate the properties and capabilities of the algorithm.

Highlights

Stiff initial-value problems (IVPs) are often solved numerically using implicit A-stable Runge-Kutta (RK) methods
There is no need to adjust the stepsize for the sake of stability, since the stability region of the method is the entire left half of the complex plane
We have designed an algorithm for determining stepsizes appropriate for stable solutions of stiff IVPs, when such solutions are computed using explicit RK methods

Summary

Introduction

Stiff initial-value problems (IVPs) are often solved numerically using implicit A-stable Runge-Kutta (RK) methods. In such methods, there is no need to adjust the stepsize for the sake of stability, since the stability region of the method is the entire left half of the complex plane. There is no need to adjust the stepsize for the sake of stability, since the stability region of the method is the entire left half of the complex plane These methods are useful when the problem is very stiff. The numerical solution will increase without bound under iteration, whereas the exact solution tends to zero This is referred to as an unstable solution, or instability with regard to stiff ODEs. It is vital to choose h at each step of the RK method so that. An algorithm for determining an appropriate stepsize h is the subject of the and subsequent sections

Theoretical Description of the Algorithm

Implementation of the Algorithm

Comments

Numerical Examples

Conclusions