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首页> 外文会议>Finite difference methods: theory and applications >Stability Properties of Repeated Richardson Extrapolation Applied Together with Some Implicit Runge-Kutta Methods
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Stability Properties of Repeated Richardson Extrapolation Applied Together with Some Implicit Runge-Kutta Methods

机译：重复Richardson外推的稳定性与某些隐式Runge-Kutta方法一起应用

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Repeated Richardson Extrapolation can successfully be used in the efforts to improve the efficiency of the numerical treatment of systems of ordinary differential equations (ODEs) mainly by increasing the accuracy of the computed results. It is assumed in this paper that Implicit Runge-Kutta Methods (IRKMs) are used in the numerical solution of systems of ODEs. If the order of accuracy of the selected IRKM is p, then the order of accuracy of its combination with the Repeated Richardson Extrapolation is at least p + 2 (assuming here that the right-hand-side of the system of ODEs is sufficiently many times continuously differentiable). However, it is additionally necessary to establish that the absolute stability properties of the new numerical methods (that are combinations of the Repeated Richardson Extrapolation and the selected IRKMs) are preserved, and this is an extremely difficult problem. Results related to the stability of the computations are derived and numerical tests with a two-parameter system of three ODEs and an atmospheric chemical scheme with 56 compounds, which is defined mathematically by a very stiff and ill-conditioned system of non-linear ODEs, are presented. The research results described in this paper can be considered as a continuation of the study carried out in Zlatev et al.: Richardson Extrapolation: Practical Aspects and Applications. De Gruyter, Berlin (2017).
机译：反复进行的Richardson外推法可以成功地用于提高常微分方程（ODE）系统数值处理效率的工作中，主要是通过提高计算结果的准确性。本文假设在ODE系统的数值解中使用了隐式Runge-Kutta方法（IRKM）。如果所选IRKM的精度等级为p，则其与重复Richardson外推法相结合的精度等级至少为p + 2（假设此处ODE系统的右侧足够多次连续微分）。但是，还必须确定要保留新数值方法（重复的Richardson外推法和选定的IRKM的组合）的绝对稳定性，这是一个非常困难的问题。得出了与计算稳定性有关的结果，并使用了由三个ODE组成的两参数系统以及由56种化合物组成的大气化学方案进行了数值测试，该方案由数学上非常僵硬且条件恶劣的非线性ODE组成，被提出。本文描述的研究结果可以看作是Zlatev等人进行的研究的延续：Richardson外推：实际方面和应用。德·格鲁伊特，柏林（2017）。

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