General linear methods for stiff differential equations. (English) Zbl 0983.65085

Butcher, J. C.

BIT 41, No. 2, 240-264 (2001).

A general class of numerical methods for stiff initial value problems that contains both the linear multistep and Runge-Kutta methods is considered. The aim of the author is to obtain particular methods that combine the low computational cost shared by the standard backward differential formula (BDF) methods of the class of multistep methods with the stability properties that possess some implicit Runge-Kutta methods.
With this idea in mind two types of methods for stiff systems on parallel and sequential implementation are considered. Then after some preliminary results on the so called inherent stability and on the order conditions, he proposes several A-stable methods with \(s\)-stages and order \(p\) where \( s=p+1\) or else \( s=p+2\) so that the implicit equations of all stages are similar to those of a diagonally implicit Runge-Kutta method and therefore its computational cost per step is equivalent to solving \(s\) times a BDF type method with different starting values.
Finally some implementation issues that include the estimation of the local truncation error, variation of step size and order for the new methods are considered.

Reviewer: Manuel Calvo (Zaragoza)

MSC:

65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
65L20	Stability and convergence of numerical methods for ordinary differential equations
65L50	Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L70	Error bounds for numerical methods for ordinary differential equations

Keywords:

initial value problems; stiff systems; general linear methods; linear multistep methods; parallel computation; error bounds; step size control; Runge-Kutta methods; backward differential formula methods; stability; order