Numerical Solution of Ordinary Differential Equations
This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consist
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Numerical Solution of Ordinary Differential Equations
Abstract This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge– Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier. In what follows we study the problem of how to solve the following initial value problem numerically: Let I ⊆ R be an interval, where I = [t0 , t0 + T ], let f : Rm × I → Rm be a function, and let x0 ∈ Rm . Note that t0 ∈ I by construction. We consider equations of the form x 0 (t) = f(x(t), t) x(t0 ) = x0 .
for all t ∈ I ,
(13.1)
Furthermore, we shall always assume that f : Rm × I → Rm is continuous and that there is a uniquely determined solution x∗ of (13.1). We begin with basic ideas and examples of methods to solve (13.1) numerically. Then we shall reformulate the initial value problem (13.1) as an operator equation. This should allow us to apply the results obtained in Chapter 12. The basic idea to solve the Problem (13.1) numerically consists in choosing a grid G of I by using the nodes t0 < t1 < · · · < tN = t0 + T and then to successively determine approximations x` of x∗ (t` ), where ` = 1, . . . , N. The resulting methods are commonly referred to as integration methods. © Springer International Publishing Switzerland 2016 W. Römisch and T. Zeugmann, Mathematical Analysis and the Mathematics of Computation, DOI 10.1007/978-3-319-42755-3_13
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13 Numerical Solution of Ordinary Differential Equations
13.1 Integration Methods for Ordinary Differential Equations To obtain an integration method, one can try different approaches. Let us start with the method given after the proof of Theorem 11.1 (cf. (11.17)), i.e., the Euler method. Since there is more than one Euler method, we refer to it here as the explicit Euler method or forward Euler method. Example 13.1 (Explicit Euler). The idea of the explicit Euler method is to replace the derivatives x 0 (t` ) in the equation x 0 (t` ) = f(x(t` ), t` ) at the nodes of the grid G by a finite difference formula for the derivative. Consequently, we set h` =df t` − t`−1 and obtain x` − x`−1 ≈ x 0 (t`−1 ) h` = f(x`−1 , t`−1 ) ,
for every ` = 1, . . . , N .
Hence, this idea directly yields the following integration method: x` = x`−1 + h` f(x`−1 , t`−1 )
for every ` = 1, . . . , N .
(13.2)
Geometrically speaking, the explicit Euler method yields a polygonal approxi
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