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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…
This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an…
In this paper, we consider a quasi-linear stochastic heat equation on $[0,1]$, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs…
We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in…
In this paper, we prove the global wellposedness of the Gross-Pitaevskii equation with white noise potential, i.e. a cubic nonlinear Schr{\"o}dinger equation with harmonic confining potential and spatial white noise multiplicative term.…
We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the…
This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts the problem reduces to stochastic…
In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing an arbitrary partial differential equation on an arbitrary lattice. An open problem is…
Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schr\"odinger equations. In particular, the Schr\"odinger-Poisson equation under homogeneous Dirichlet boundary…
We consider higher order viscous Burgers' equations with generalized nonlinearity and study the associated initial value problems for given data in the $L^2$-based Sobolev spaces. We introduce appropriate time weighted spaces to derive…
In this paper we study parabolic stochastic partial differential equations defined on arbitrary bounded domain $\cO \subset \bR^d$ allowing Hardy inequality: $$ \int_{\cO}|\rho^{-1}g|^2\,dx\leq C\int_{\cO}|g_x|^2 dx, \quad \forall g\in…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as…
We study the KPZ equation with a $1+1-$dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of \cite{bqs}, i.e., the variance of the solution at time $t$ is of order $t^{2/3}$. Instead of…
This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically,…
We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain…
We prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain.…
Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated It\^o stochastic process (with zero mean) obtained from data which is taken in…
In this paper, we establish a central limit theorem (CLT) and the moderate deviation principles (MDP) for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results…
We treat some classes of linear and semilinear stochastic partial differential equations of Schr\"odinger type on $\mathbb{R}^d$, involving a non-flat Laplacian, within the framework of white noise analysis, combined with Wiener-It\^o chaos…