6. Numerical Algorithms

An important class of discrete dynamical systems arises when continuous dynamical systems are discretized to be solved numerically on a computer. The main concern is to what extent the numerical discrete solution mimics or resembles the solution of the original continuous system. This chapter describes some basic properties of the algorithms that are used to obtain numerical solutions for ODEs and PDEs. It is an abridgment of a chapter in Müller and von Storch (2004). Numerical algorithms executed on a digital computer have to introduce a finite time step, and a finite grid size in case of PDEs. Numerical algorithms introduce two kind of basic errors: round-off errors and truncation errors. Round-off errors occur since digital computers have only a finite word length, a finite number of bits to represent real numbers. Round-off errors are property of the machine on which the numerical code is executed. Truncation errors occur when the derivative of a function is replaced by a finite difference approximation. Truncation errors are under the control of the modeler. They can be reduced by decreasing the time step or grid spacing.