随机波动方程的变步长Runge-Kutta方法
Variable stepsize Runge-Kutta methods for stochastic wave equations
摘要：
我们表明，现有的龙格-库塔方法的常微分方程（ODS）可以修改解决随机微分方程（SDES）的强解，提供适当的变化是选择步长的方式。结果sde方案的阶数是ode方案阶数的一半。具体而言，我们表明，显式的9阶龙格库塔法（与嵌入式8阶方法）的ODES产生一个4.5阶的SDES方法，可以实现可变步长。该方法是通过求解系统的随机波动方程所产生的主方程和多体薛定谔方程的SDES测试。
英文摘要：
We show that existing Runge-Kutta methods for ordinary differential equations (odes) can be modified to solve stochastic differential equations (sdes) with strong solutions provided that appropriate changes are made to the way stepsizes are selected. The order of the resulting sde scheme is half the order of the ode scheme. Specifically, we show that an explicit 9th order Runge-Kutta method (with an embedded 8th order method) for odes yields an order 4.5 method for sdes which can be implemented with variable stepsizes. This method is tested by solving systems of sdes originating from stochastic wave equations arising from master equations and the many-body Schroedinger equation.
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