The parallel waveform relaxation stochastic Runge–Kutta method for stochastic differential equations

For large-scale non-autonomous Stratonovich stochastic differential equations, we study a very general parallel waveform relaxation process which is on the basis of stochastic Runge–Kutta (SRK) method of mean-square order 1.0 in this literature. The convergence of the whole parallel numerical iterative scheme can be guaranteed and the scheme provides better properties in terms of decreasing the load of the computation and operating speed. At the same time, the related limit method is also introduced as the continuous approximation derived from the iterative scheme. In the approximation interval, it is worth noting that the mean-square order of the parallel numerical iterative scheme can be kept consistent with the previous SRK method at any arbitrary time point, not just at discrete points. Some numerical simulations are presented to elaborate the computing efficiency of the parallel numerical iterative scheme.