Galerkin Spectral Method of Stochastic Partial Differential Equations Driven by Multivariate Poisson Measure

A considerable body of prior research has been dedicated to devising efficient and high-order numerical methods for solving stochastic partial differential equations (SPDEs) driven by discrete or continuous random variables. The majority of these efforts have concentrated on introducing numerical approaches for SPDEs involving independent random variables, with limited exploration into solving SPDEs driven by dependent and discrete random variables. In this paper, we introduce a Galerkin spectral method tailored for addressing SPDEs driven by multivariate Poisson variables. Our novel numerical method is applied to solve the one- and two-dimensional stochastic Poisson equations, as well as the one and two-dimensional stochastic heat equation, stochastic KdV equation, and stochastic Burgers equation, respectively. Notably, all the presented stochastic equations are governed by two dependent Poisson random variables.