This paper studies the long time stability of both stochastic heat equations on a bounded domain driven by a correlated noise and their approximations. It is popular for researchers to prove the intermittency of the solution which means that the moments of solution to stochastic heat equation usually grow exponentially to infinite and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on bounded domain. We also present numerical experiments which are consistent with our theoretical results.

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIR model. For the real significance of the numerical analysis on stochastic SIR model, a linearly implicit Euler method with truncated Wiener process is introduced. The numerical positivity is obtained by the truncated Wiener process, which is the basis for the investigation of convergence and dynamical behavior. The numerical dynamical behavior is obtained by an exponential presentation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIR model.

In this paper, the numerical positivity and almost surely exponential stability of the stochastic heat equation are discussed. The finite difference method and the split-step backward Euler are considered for spatial and temporal, respectively. Motivated from physical applications such as temperature, finance and so on, positivity has real significance, which is volatilized by some common numerical treatments. To this end the numerical positivity is obtained by the properties of M -matrix and the truncated random variables, which overcomes the unboundedness of the random variables. For the investigation of the almost surely exponential stability, a stochastic stability matrix is introduced, and then the stability analysis reduces to the estimation of the eigenvalues and martingales thorough a family of matrices and perturbation theorems. The stabilization ability of the multiplicative noise is verified again from a generalization of the stochastic heat equation. Finally, some numerical experiments are given to validate our numerical results.

The paper deals with the asymptotical mean-square stability of the linear theta-methods under variable stepsize and transformation approach for stochastic pantograph differential equations. A limiting equation for the analysis of numerical stability is introduced by Kronecker products. Under the condition which guarantee the stability of exact solutions, the optimal stability region of the linear theta-methods under variable stepsize is given by using the limiting equation, i.e. theta is an element of (1/2, 1], which is the same to the deterministic problems. Moreover, the linear theta-methods under the transformation approach are also considered and the result of the stability is improved for theta = 1/2 Finally, numerical examples are given to illustrate the asymptotical mean-square stability under variable stepsize and transformation approach.

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIS model. Compared with the existing numerical methods, a linearly backward Euler method with truncated Wiener process is introduced with a less computational cost and a better dynamic behavior. We discuss the numerical positivity by the truncated Wiener process, which is the basis for the investigation of convergence and dynamic behavior. The numerical dynamical behaviors (extinction and persistence) are obtained by an exponential representation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIS model.

The paper deals with the numerical asymptotical mean-square stability of split-step ? methods for stochastic pantograph differential equations, which is the generalization of deterministic pantograph equations. Instead of the quasi-geometric mesh, a fully-geometric mesh, widely used for deterministic problems, is employed. A useful technique, the limiting equation, for deterministic problems is also extended to stochastic problems based on Kronecker product. Under the exact stability condition, the stability region of the split-step ? methods is discussed, which is an improvement of some existing results. Moreover, such technique is also available to stochastic pantograph differential equations with Poisson jumps. Meanwhile, compared with the destabilization of Wiener process, the stabilization of Poisson jumps is replicated by numerical processes. Finally, numerical examples are given to illustrate that our numerical stability condition is nearly necessary for stochastic problems.