Numerical methods for the solution of ordinary differential equations: initial value problems.
Abstract:ESDU 86011 considers methods for the numerical evaluation of initial value problems in ordinary differential equations. Some of the methods are applicable to a set of n first-order differential equations with an appropriate set of n initial conditions specified at a single value of the independent variable. It is also shown that a differential equation of higher order than one, provided it is possible to isolate the highest derivative of the dependent variable, can be treated using the same methods. Other methods apply to a set of m second-order equations with an appropriate set of 2 x m initial conditions also specified at a single value of the independent variable. Two broad classes of methods are considered; single-step methods, specifically those that are referred to as Runge-Kutta and acceleration methods, and those multi-step methods that fall within the term predictor-corrector. For the Runge-Kutta class of formulae only fourth-order methods are described. The schemes presented provide alternative ways of achieving reduction in either truncation error or computation time. Adaptive control of the step size is also discussed. The acceleration methods, the Wilson-theta Linear Acceleration Method and the Newmark Generalised Acceleration Method, are described. For the predictor-corrector class of formulae general variable-order equations are presented. General remarks concerning stability and stiff equations are included. Not included are those methods that are appropriate for boundary value problems, partial differential equations and methods that deal directly with differential equations that are higher than second order.
|Data Item ESDU 86011|