Application Of High Order Multisymplectic Splitting Scheme To The Deterministic And Stochastic Partial Differential Equations

In this article, we propose a high order multisymplectic splitting scheme to solve a class of deterministic and stochastic Hamilton system functions and it has an immeasurable role. It is a very effective composite finite difference method, which is combine with multisymplectic algorithm、high order difference operator and splitting method. We apply this scheme to the numerical calculation of the deterministic nonlinear Schrodinger equation and stochastic nonlinear Schrodinger equation, and we can find the superiority of this scheme through the numerical simulation of the solitary wave and the collision of two waves. In simple terms, firstly, to the stochastic nonlinear Schrodinger equation and deterministic nonlinear Schrodinger equation, we respectively give the corresponding Hamilton multisymplectic constructions; secondly, dividing the original problem into a linear sub-problem and one or two nonlinear sub-problems through the two-step splitting scheme and three-step splitting scheme, then the corresponding two-step multisymplectic splitting construction and three-step multisymplectic splitting construction are given; thirdly, we apply the fourth-order difference scheme to solve the linear problem, while giving the corresponding numerical explicit expression to the nonlinear problem; finally, to the stochastic nonlinear Schrodinger equation, the discrete charge conservation law and the recursive relation of energy are given, and to the deterministic nonlinear Schrodinger equation, the corresponding discrete charge conservation law and energy conservation law are given.In the last part of this paper, the corresponding numerical simulation experiments are given and they prove the convergence of the algorithm is the fourth-order accuracy, and observe the propagation of the solitary wave and collision of waves with different random coefficients. In addition, we use the data which are obtained by numerical experiments to show the charge conservation law and the recursive relation of energy, which match the theoretical results.