Numerical Simulation of ODE Models

Abstract

In the preceding chapter we had worked out how to establish possibly large ODE models for systems biological networks. In the present chapter, we deal with their numerical simulation. For this purpose, we describe various numerical integrators for initial value problems in necessary detail. In Sect. 2.1, we present basic concepts to characterize different discretization methods. We start with local versus global discretization errors, first in theory, then in algorithmic realization. Stability concepts for discretizations lead to an elementary pragmatic understanding of the term “stiffness” of ODE systems. In the remaining part of the chapter, different families of integrators such as Runge-Kutta methods, extrapolation methods, and multistep methods are characterized. From a practical point of view they are divided into explicit methods (Sect. 2.2), implicit methods (Sect. 2.3), and linearly implicit methods (Sect. 2.4), to be discussed in terms of their structural strengths and weaknesses. Finally, in Sect. 2.5, a roadmap of numerical methods is given together with two moderate problems that look rather similar, but require different numerical integrators. Moreover, we present a more elaborate example concerning the dynamics of tumor cells; therein we show, what kind of algorithmic decisions may influence the speed of computations.