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Backward error analysis for multistep methods. (English) Zbl 0941.65077
Recently, much insight into the numerical solution of ordinary differential equations by one-step methods has been obtained via a backward error analysis. The idea of the backward error analysis consists in searching for a modified differential equation \[ \tilde{y}'=f(\tilde{y})+h f_2(\tilde{y})+h^2 f_3(\tilde{y})+\cdots, \tag{1} \] whose exact solution is identical to the numerical solution, \(y_n=\tilde{y}(n h),\) where \(y_0, y_1, y_2, \ldots\) is a numerical solution of the initial value problem \[ y'=f(y), \quad y(0)=y_0 \tag{2} \] approximating \(y(x)\) at equidistant grid points. The purpose of this paper is to extend the formal and the rigorous backward analysis to multistep methods. The exposition starts with examination of some numerical experiments illustrating the good agreement of the numerical approximation of (2) with the solution of (1). This is followed by the construction of the modified differential equation for multistep methods based on a Lie derivative of B-series and the discussion of rigorous error estimates for a particular choice of the starting values \(y_0, y_1, y_2, \ldots, y_{k-1}.\)
Then a complete backward error analysis for multistep methods is presented. The numerical solution is written as the exact solution of the principal modified equation superposed by oscillating terms which are defined as the solution of so-called parasitic modified equations. Finally, it is explained how this theory can be applied for the study of the long-time behaviour of multistep solutions.

MSC:
65L70 Error bounds for numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
65L05 Numerical methods for initial value problems
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