论文标题
刚性随机微分方程的显式稳定多胎方法
Explicit stabilized multirate method for stiff stochastic differential equations
论文作者
论文摘要
由于其扩展的稳定性域，稳定的显式方法对于大型刚性随机微分方程（SDE）的大型系统特别有效。但是，当很少有“快速”的自由度引起严重的刚度时，它们会散发效率，因为同时评估了刚性和非固定项。因此，受[A. Abdulle，M。J。Grote和G. Rosilho de Souza，Preprint（2020），Arxiv：2006.00744]我们引入了一个随机修改方程，其刚度仅取决于“慢速”项。通过将这种修改的方程式与稳定的显式方案整合在一起，我们设计了一种多次方法，该方法克服了由几个严重刚性的术语引起的瓶颈，并恢复了大型非线性SDE系统的稳定方案的效率。该方案不是基于SDE的任何规模分离假设，因此它是由于随机抛物线偏微分方程在局部精制的网格上的空间离散化所引起的问题。多胎方案具有强大的顺序1/2，弱顺序1，并且在模型问题上证明了其稳定性。数值实验证实了该方案的效率和准确性。
Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few "fast" degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the "slow" terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.
