2019/07/05 by Chuchu Chen, Jialin Hong, Chen, Chuchu +3
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1907.02825
openalex publication_date 2019/07/05 · openalex created_date 2019/07/12 · openalex updated_date 2026/07/28
We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation, which is a perturbation of the Wong--Zakai approximation of the rough differential equation. For a symplectic method applied to a rough Hamiltonian system, the associated stochastic modified equation is proved to have a Hamiltonian formulation. Second, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in the rough path theory. Third, if increments of noises are simulated by truncated random variables, we show that the one-step error can be made exponentially small with respect to the time step size. Numerical experiments verify our theoretical results.