The content discusses the development of new numerical integration methods based on linear combinations of compositions of a basic second-order scheme. The key points are:
Extrapolation methods are a class of efficient numerical integrators for initial value problems, especially when high accuracy is desired. They are constructed by taking linear combinations of compositions of a basic second-order scheme.
The authors propose a more general family of integrators that can be considered as a generalization of extrapolation methods. These methods involve linear combinations of compositions of the basic second-order scheme with additional coefficients.
The additional coefficients can be used to increase the order of preservation of qualitative properties of the original differential equation, such as symplecticity or unitarity, or to reduce the most significant contributions to the truncation error.
The authors provide a general analysis of these generalized extrapolation methods and construct new schemes of orders 4, 6, and 8. These new methods are shown to be more efficient than standard extrapolation methods in some cases.
The authors also analyze the latency problem that can arise when implementing these methods in a parallel environment. They show how the methods can be modified to reduce the latency by delaying the summation of the different compositions.
Numerical experiments on the Kepler problem and the Lotka-Volterra system demonstrate the improved performance of the new schemes compared to standard extrapolation methods.
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arxiv.org
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