Abstract
Explicit numerical methods have a great advantage in computational cost, but they usually fail to preserve the conserved quantity of original stochastic differential equations (SDEs). In order to overcome this problem, two improved versions of explicit stochastic Runge–Kutta methods are given such that the improved methods can preserve conserved quantity of the original SDEs in Stratonovich sense. In addition, in order to deal with SDEs with multiple conserved quantities, a strategy is represented so that the improved methods can preserve multiple conserved quantities. The mean-square convergence and ability to preserve conserved quantity of the proposed methods are proved. Numerical experiments are implemented to support the theoretical results.
Highlights
Stochastic differential equations (SDEs) model the process in practical applications more realistic by taking stochastic effects into consideration
From this point of view, and based on the fact that explicit numerical methods are superior with respect to computational cost, we study the improved version of explicit stochastic Runge–Kutta (ESRK) methods such that the conserved quantities are preserved as accurately as possible
In order to further improve the accuracy of simulation, we introduce an improved ESRK method based on the optimization technique
Summary
Stochastic differential equations (SDEs) model the process in practical applications more realistic by taking stochastic effects into consideration. Some implicit numerical methods can be proved to preserve the conserved quantity of original SDEs exactly, the errors caused by computation of implicit nonlinear equations make the solutions of these nonlinear equations approximate, rather than exact. In the process of constructing numerical methods, it is not necessary for us to make sure that the numerical solutions preserve the conserved quantity exactly, instead, we only need to make it as close to conserved quantity of original SDEs as possible From this point of view, and based on the fact that explicit numerical methods are superior with respect to computational cost, we study the improved version of explicit stochastic Runge–Kutta (ESRK) methods such that the conserved quantities are preserved as accurately as possible.
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