A high-order discontinuous Galerkin method for Itô stochastic ordinary differential equations

Abstract

In this paper, we develop a high-order discontinuous Galerkin (DG) method for strong solution of Itô stochastic ordinary differential equations (SDEs) driven by one-dimensional Wiener processes. Motivated by the DG method for deterministic ordinary differential equations (ODEs), we first construct an approximate deterministic ODE with a random coefficient on each element using the well-known Wong–Zakai approximation theorem. Since the resulting ODE converges to the solution of the corresponding Stratonovich SDE, we apply a transformation to the drift term to obtain a deterministic ODE which converges to the solution of the original SDE. The corrected equation is then discretized using the standard DG method for deterministic ODEs. We prove that the proposed stochastic DG (SDG) method is equivalent to an implicit stochastic Runge–Kutta method. Then, we study the numerical stability of the SDG scheme applied to linear SDEs with an additive noise term. The method is shown to be numerically stable in the mean sense and also A-stable. As a result, it is suitable for solving stiff SDEs. Moreover, the method is proved to be convergent in the mean-square sense. Numerical evidence demonstrates that our proposed DG scheme for SDEs with additive noise has a strong convergence order of 2p+1, when p-degree piecewise polynomials are used. When applied to SDEs with multiplicative noise, the SDG method is strongly convergent with order p. Several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method. In particular, we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems.