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Backward error analysis for numerical integrators. (English) Zbl 0935.65142

The author considers the relationship between solutions to a given system of ordinary differential equations (vector fields) \(\frac {d}{dt}\mathbf x = \mathbf Z(\mathbf x)\), numerical approximations \(\mathbf x_{n+1}={\mathbf\Psi}_{\delta t}(\mathbf x_n)\) to them, and solutions to associated modified equations \(\frac {d}{dt}\mathbf x = \widetilde{\mathbf X}_i(\mathbf x; \delta t)\), \((i\geq 1).\) The author revisits backward error analysis by using a simple recursive scheme for the definition of the modified vector fields \(\widetilde{\mathbf X}_i(\delta t)\) that does not require explicit Taylor series expansion of the numerical method and the corresponding flow maps. As an application it is discussed the long integration of chaotic Hamiltonian systems and the approximation of time averages along numerically trajector.
Reviewer: K.Najzar (Praha)

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65P20 Numerical chaos
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
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