Stochastic conformal symplectic exponential Runge–Kutta–Nyström integrators for solving damped second-order SDEs with applications in damped stochastic nonlinear wave equations

Abstract

In this paper, we propose implicit and explicit stochastic conformal symplectic exponential Runge–Kutta–Nyström (SCSE-RKN) integrators to solve second-order stochastic differential equations (SDEs) with constant or time-dependent damping. Under certain conditions on coefficient functions, the conformal quadratic invariants of damped second-order SDEs exist. Furthermore, the corresponding damped stochastic Hamiltonian systems are conformal symplectic. We theoretically and experimentally demonstrate that the implicit and explicit SCSE-RKN integrators preserve the conformal quadratic invariant and conformal symplectic structure accurately. Additionally, we establish the existence and uniqueness of numerical solution for the implicit SCSE-RKN integrator. And we provide a detailed analysis of the mean-square convergence orders for both implicit and explicit SCSE-RKN integrators. In comparison to other classical methods, our proposed integrators exhibit superior structure-preserving performance, higher accuracy, and better efficiency. Using the SCSE-RKN integrators in the time direction, we achieve excellent numerical results for two renowned damped stochastic nonlinear wave equations.