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On the numerical integration of ordinary differential equations by processed methods. (English) Zbl 1079.65075

A one-step integrator \(\psi_h: \mathbb{R}^D\to \mathbb{R}^D\) with time step \(h\) for an ordinal differential equation \(x'= f(x)\), \(f: \mathbb{R}^D\to \mathbb{R}^D\), can be enhanced by “processing” based on a postprocessors \(\pi_h: \mathbb{R}^D\to \mathbb{R}^D\), to obtain a new integrator \(\widehat\psi_h:=\pi_h\circ\psi_h\circ \pi^{-1}_h\) [see J. C. Butcher, Conf. numer. Solut. differential Equat., Dundee/Scotland 1969, 133–139 (1969; Zbl 0185.41301) and, e.g., R. I. McLachlan and G. R. W. Quispel, Acta numerica 11, 341–434 (2002)].
The present paper develops a general theory of processing; in particular, the authors analyze the effective order conditions for a given integrator \(\psi_h\) and then discuss optimal processor \(\pi_h\) and feasible “cheap” approximations \(\widehat\pi_h\). The theoretical analysis is illustrated in detail by two numerical examples.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
22E60 Lie algebras of Lie groups
34A45 Theoretical approximation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Citations:

Zbl 0185.41301
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