Abstract
We review some methods for high precision time integration: it is not easy to ensure stability, precision and numerical efficiency at the same time. Operator splitting—when it works—can be a good way to satisfy all these constraints; in some cases, the order of the splitting schemes can be enhanced by extrapolation; nevertheless, the applicability of splitting is limited due to non commutativity. As an alternative to splitting, we introduce preconditioned Runge–Kutta (PRK) schemes: the preconditioning is included in the scheme, instead of being put aside for implementation. Examples of PRK schemes are given including the extrapolation of the residual smoothing scheme, and sufficient conditions for stability are described.
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Schatzman, M. Toward Non Commutative Numerical Analysis: High Order Integration in Time. Journal of Scientific Computing 17, 99–116 (2002). https://doi.org/10.1023/A:1015140328635
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DOI: https://doi.org/10.1023/A:1015140328635