Time Discrete Approximation of Deterministic Differential Equations
In this chapter we summarize the basic concepts and assertions of the numerical analysis of initial value problems for deterministic ordinary differential equations. The material is presented so as to facilitate generalizations to the stochastic setting a
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Time Discrete Approximation of Deterministic Differential Equations In this chapter we summarize the basic concepts and assertions of the numerical analysis of initial value problems for deterministic ordinary differential equations. The material is presented so as to facilitate generalizations to the stochastic setting and to highlight the differences between the deterministic and stochastic cases.
8.1
Introduction
In general it is not possible to find explicitly the solution x = x(t; to, xo) of an initial value problem (IVP)
(1.1)
. x
( ) = dx dt = a t,x ,
x(to) = Xo
for the deterministic differential equations that occur in many scientific and technological models. Even when such a solution can be found, it may be only in implicit form or too complicated to visualize and evaluate numerically. Necessity has thus lead to the development of methods for calculating numerical approximations to the solutions of such initial value problems. The most widely applicable and commonly used of these are the time discrete approximation or difference methods, in which the continuous time differential equation is replaced by a discrete- time difference equation generating values Y1, Y2, ... , Yn, ... to approximate X(t1; to, xo), X(t2; to, xo), ... , x(t n ; to, xo), ... at given discretization times to < t1 < t2 < ... < tn < .... These approximations should be quite accurate, one hopes, if the time increments Ll n = tn+1 - tn for n = 0, 1, 2, ... are sufficiently small. As a background for the development of discretization methods for stochastic differential equations, in this chapter we shall review the basic difference methods used for ordinary differential equations and consider some related issues such as their convergence and stability. The simplest difference method for the IVP (1.1) is the Euler method (1.2) for a given time discretization to
< t1 < t2 < ... < tn < ... with increments Ll n
=tn+1 -tn where n = 0, 1,2, .... Once the initial value Yo has been specified, usually Yo = Xo, the approximations Yb Y2, ... , Yn, ... can be calculated by recursively applying formula (1.2). We can derive (1.2) by freezing the right P. E. Kloeden et al., Numerical Solution of Stochastic Differential Equations © Springer-Verlag Berlin Heidelberg 1992
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CHAPTER 8. DETERMINISTIC DIFFERENTIAL EQUATIONS
hand side of the differential equation over the time interval tn :::; t < tn+! at the value a(tn, Yn) and then integrating to obtain the tangent to the solution z(tjtn,Yn) of the differential equation with the initial value z(tn) = Yn. The difference (1.3) which is generally not zero, is called the local discretization error for the nth time step. This is usually not the same as the global discretization error (1.4) for the same time step, which is the error with respect to the sought solution of the original IVP (1.1). Nevertheless, we can use the local discretization error to estimate the global discretization error. It must be emphasized that (1.3) and (1.4) assume that we can perform all arithmetic calculations exactly
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