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

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Pade approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the averaged energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.

Absolute continuity and numerical approximation of stochastic Cahn--Hilliard equation with unbounded noise diffusion

In this article, we develop and analyze a full discretization, based on the spatial spectral Galerkin method and the temporal drift implicit Euler scheme, for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise. By introducing an appropriate decomposition of the numerical approximation, we first use the factorization method to deduce the a priori estimate and regularity estimate of the proposed full discretization. With the help of the variation approach, we then obtain the sharp spatial and temporal convergence rate in negative Sobolev space in mean square sense. Furthermore, the sharp mean square convergence rates in both time and space are derived via the Sobolev interpolation inequality and semigroup theory. To the best of our knowledge, this is the first result on the convergence rate of temporally and fully discrete numerical methods for the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise.

Energy-preserving multi-symplectic Runge-Kutta methods for Hamiltonian wave equations

It is well-known that a numerical method which is at the same time geometric structure-preserving and physical property-preserving cannot exist in general for Hamiltonian partial differential equations. In this paper, we present a novel class of parametric multi-symplectic Runge-Kutta methods for Hamiltonian wave equations, which can also conserve energy simultaneously in a weaker sense with a suitable parameter. The existence of such a parameter, which enforces the energy-preserving property, is proved under certain assumptions on the fixed step sizes and the fixed initial condition. We compare the proposed method with the classical multi-symplectic Runge-Kutta method in numerical experiments, which shows the remarkable energy-preserving property of the proposed method and illustrate the validity of theoretical results.

Stochastic modified equations for symplectic methods applied to rough Hamiltonian systems based on the Wong--Zakai approximation

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.

Wellposedness and regularity estimate for stochastic Cahn--Hilliard equation with unbounded noise diffusion

In this article, we consider the stochastic Cahn--Hilliard equation driven by multiplicative space-time white noise with diffusion coefficient of sublinear growth. By introducing the spectral Galerkin method, we first obtain the well-posedness of the approximated equation in finite dimension. Then with the help of the semigroup theory and the factorization method, the approximation processes is shown to possess many desirable properties. Further, we show that the approximation process is strongly convergent in certain Banach space via the interpolation inequality and variational approach. Finally, the global existence and regularity estimate of the unique solution process are proven by means of the strong convergence of the approximation process.

Approximation of Invariant Measures for Stochastic Differential Equations with Piecewise Continuous Arguments via Backward Euler Method

For the stochastic differential equation (SDE) which has piecewise continuous arguments (PCAs), is driven by multiplicative noises and its drift coefficients are dissipative, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to inherit numerically the invariant measure of SDE with PCAs, we apply the backward Euler (BE) method to the equation, and prove that the numerical solution at integer time is not only Markovian but also reproduces a unique numerical invariant measure. We present the time-independent weak error analysis for the method under certain hypothesis. Further, we show that the numerical invariant measure converges to the original one with order 1. Numerical experiments verify the theoretical analysis.

The probabilistic superiority of stochastic symplectic methods via large deviations principles

It is well known that symplectic methods have been rigorously shown to be superior to non-symplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the probabilistic superiority of stochastic symplectic methods by means of the theory of large deviations. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Concerning that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gartner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but non-symplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the hitting probability of the mean position and mean velocity of the stochastic oscillator. To the best of our knowledge, this is the first result about using large deviations principle to show the superiority of stochastic symplectic methods compared with non-symplectic ones in the existing literature.

Convergence in Density of Splitting AVF Scheme for Stochastic Langevin Equation

In this article, we study the density function of the numerical solution of the splitting averaged vector field (AVF) scheme for the stochastic Langevin equation. To deal with the non-globally monotone coefficient in the considered equation, we first present the exponential integrability properties of the exact and numerical solutions. Then we show the existence and smoothness of the density function of the numerical solution by proving its uniform non-degeneracy in Malliavin sense. In order to analyze the approximate error between the density function of the exact solution and that of the numerical solution, we derive the optimal strong convergence rate in every Malliavin--Sobolev norm of the numerical scheme via Malliavin calculus. Combining the approximation result of Donskers delta function and the smoothness of the density functions, we prove that the convergence rate in density coincides with the optimal strong convergence rate of the numerical scheme.

Exponential Integrators for Stochastic Maxwells Equations Driven by It^o Noise

This article presents explicit exponential integrators for stochastic Maxwells equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is $frac 12$ for general multiplicative noise. Combing a proper decomposition with the stochastic Fubinis theorem, the strong order of the proposed scheme is shown to be $1$ for additive noise. Moreover, for linear stochastic Maxwells equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.

Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise

In this article, we consider the stochastic Cahn--Hilliard equation driven by space-time white noise. We discretize this equation by using a spatial spectral Galerkin method and a temporal accelerated implicit Euler method. The optimal regularity properties and uniform moment bounds of the exact and numerical solutions are shown. Then we prove that the proposed numerical method is strongly convergent with the sharp convergence rate in a negative Sobolev space. By using an interpolation approach, we deduce the spatial optimal convergence rate and the temporal super-convergence rate of the proposed numerical method in strong convergence sense. To the best of our knowledge, this is the first result on the strong convergence rates of numerical methods for the stochastic Cahn--Hilliard equation driven by space-time white noise. This interpolation approach is also applied to the general noise and high dimension cases, and strong convergence rate results of the proposed scheme are given.