B-Series Analysis of Stochastic Runge-Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values
B-Series Analysis of Stochastic Runge-Kutta Methods That Use an Iterative Scheme to Compute Their Internal Stage Values
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DOI：
10.1137/070704307
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发表时间：
2008-10
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影响因子：
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通讯作者：
K. Debrabant;Anne Kværnø
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文献类型：
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作者：
K. Debrabant;Anne Kværnø
近年来，隐式随机Runge-Kutta(SRK)方法在强逼近和弱逼近两方面都得到了发展。对于这些方法，阶段值仅隐式给出。然而，在实际应用中，这些隐式方程的求解通常采用简单迭代、修正牛顿迭代或全牛顿迭代等迭代格式。我们使用一种构造随机B级数的统一方法来分析迭代Runge-Kutta方法的阶次，这种方法对随机微分方程和Stratonovich随机微分方程都有效，对弱收敛和强收敛都适用。此外，本文中应用的分析技术也可用于许多其他类似的情况。
In recent years, implicit stochastic Runge-Kutta (SRK) methods have been developed both for strong and weak approximations. For these methods, the stage values are only given implicitly. However, in practice these implicit equations are solved by iterative schemes such as simple iteration, modified Newton iteration or full Newton iteration. We employ a unifying approach for the construction of stochastic B-series which is valid both for It o- and Stratonovich-stochastic differential equations (SDEs) and applicable both for weak and strong convergence to analyze the order of the iterated Runge-Kutta method. Moreover, the analytical techniques applied in this paper can be of use in many other similar contexts.