The Principles of Numerical Modeling

Abstract

The solution of a differential equation can be obtained by mathematical operations which involve integration. In order to obtain a solution using a digital computer the analytic processes of integration must be replaced by a numerical method which can yield an approximation to the true solution. A continuous variable x which is a function of the independent variable t (i.e. time) is represented within a digital simulation by a series of numbers, or samples, x ₀, x ₁, x ₂,…, x _n . These samples define the variable x in terms both of magnitude and sign at the times t ₀, t ₁, t ₂,…, t _n . For many purposes it may be assumed that these samples are equally spaced in time and it is clear that if the sampling rate is high the information loss owing to sampling can be small. However, the step size, h, does affect the size of the error in the numerical approximation and the choice of h must depend upon the dynamic characteristics of the system under consideration.