Strong convergence of a Verlet integrator for the semi-linear stochastic wave equation

Abstract

The full discretization of the semi-linear stochastic wave equation is considered. The discontinuous Galerkin finite element method is used in space and analyzed in a semigroup framework, and an explicit stochastic position Verlet scheme is used for the temporal approximation. We study the stability under a CFL condition and prove optimal strong convergence rates of the fully discrete scheme. Numerical experiments illustrate our theoretical results. Further, we analyze and bound the expected energy and numerically show excellent agreement with the energy of the exact solution.

Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

“…The discontinuous Galerkin finite element method is flexible to deal with the complex computational domain and is easy to construct locally high-order approximations, which has been extensively studied in [2,3,17]. In this subsection, we discretize (1.1) with Dirichlet boundary condition by using the interior penalty discontinuous Galerkin finite element method in space.…”

“…The nonlinear stochastic wave equations play an important role in a wide range of applications in the field of engineering, science, etc., and are commonly used to describe a variety of physical processes, such as the motion of a strand of DNA in a liquid, the motion of shock waves on the surface of the sun, the dynamics of the primary current density vector field within the grey matter of the human brain and the sound propagation in the sea and so on (see e.g., [3,7,8,11] and references therein). In this paper, we consider the following nonlinear stochastic wave equation driven by multiplicative noise…”

“…Especially, numerical schemes preserving energy evolution law often yield physically correct results and numerical stability in computation. Much effort has been devoted to the numerical method of stochastic wave equations (see [1,3,6,8,9,10,16,21,26] and references therein). For instance, [8] proves that the fully-discrete scheme based on finite element method and stochastic trigonometric scheme preserves the linear growth of the averaged energy for the linear stochastic wave equation with additive noise.…”

“…For instance, [8] proves that the fully-discrete scheme based on finite element method and stochastic trigonometric scheme preserves the linear growth of the averaged energy for the linear stochastic wave equation with additive noise. With regard to the nonlinear stochastic wave equation with additive noise, [3] shows that the discontinuous Galerkin finite element method satisfies the trace formula. [10] proposes a full discretization by adopting spectral Galerkin method and averaged vector field method preserving the averaged energy evolution law of the stochastic cubic wave equation with additive noise.…”