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On the construction of symmetric second order methods for ODEs

APPLIED MATHEMATICS LETTERS 2019年 98卷 41-48页

作者： Blanes, Sergio Univ Politecn Valencia Inst Univ Matemat Multidisciplinar Valencia Spain

We show how to build explicit symmetric second order methods for solving ordinary differential equations. These methods are very useful when low accuracy is required or when higher order ones by extrapolation or composition are desired to reach high accuracy. The proposed schemes are obtained by using simple splitting methods on an extended phase space. By construction, the schemes are symmetric and of second order allowing to recover most well known and frequently used schemes from the literature. This provides a simple proof on their time symmetric structure that is very useful when the schemes are used to get higher order methods by extrapolation or composition. We show how to obtain them in the general case as well as how to get Nystrom methods, methods for stiff problems, Lie group integrators or symplectic integrators, but the technique can also be used to build explicit and implicit methods for many other problems. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Numerical methods for ODEs symmetric second order methods Extrapolation Composition methods

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Novel parallel in time integrators for ODEs

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APPLIED MATHEMATICS LETTERS 2021年 122卷 107542-107542页

作者： Blanes, Sergio Univ Politecn Valencia Inst Univ Matemat Multidisciplinar Valencia Spain

We present a novel class of integrators for differential equations that are suitable for parallel in time computation, whose structure can be considered as a generalization of the extrapolation methods. Starting with a low order integrator (preferably a symmetric second order one) we can build a set of second order schemes by few compositions of this basic scheme that can be computed in parallel. Then, a proper linear combination of the results (obtained from the order conditions associated to the corresponding Lie algebra) allows us to obtain new higher order methods. In this letter we present the structure of the methods, how to obtain several methods, we notice some order barriers that depend on the structure of the compositions used and finally, we show how this analysis can be further carried to obtain new and higher order schemes. (C) 2021 Elsevier Ltd. All rights reserved.

关键词： Numerical methods for ODEs Parallel in time methods symmetric second order methods Extrapolation Composition methods

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