Analysis of splitting schemes for the stochastic Allen-Cahn equation

Charles-Edouard Bréhier (CNRS & Université Lyon 1), Jianbao Cui, Ludovic Goudenège, Jialin Hong

The stochastic Allen-Cahn equation, with additive space-time white noise perturbation, in dimension \(1\), is given by the following semilinear SPDE \[dX(t)=AX(t)dt+\bigl(X(t)-X(t)^3\bigr)dt+dW(t).\] Since the nonlinearity \(x\mapsto x-x^3\) is not globally Lipschitz continuous, the design of suitable temporal discretization scheme is delicate. We propose to use a splitting strategy, taking into account that the flow \(\bigl(\Phi_t(z)\bigr)_{t\ge 0}\) of the ODE \(\dot{z}=z-z^3\) is exactly known.

We study numerical schemes defined as \[X_{n+1}=e^{\Delta t A}\Phi_{\Delta t}(X_n)+\int_{n\Delta t}^{(n+1)\Delta t}e^{(n\Delta t-t)A}dW(t),\] (exact sampling of the stochastic convolution), or as \[X_{n+1}=S_{\Delta t}\Phi_{\Delta t}(X_n)+S_{\Delta t}\bigl(W((n+1)\Delta t)-W(n\Delta t)\bigr)\] with \(S_{\Delta t}=(I-\Delta tA)^{-1}\) (semi-implicit discretization of the stochastic convolution).

Moment estimates, as well as strong and weak convergence rates, will be presented.

I will also present numerical simulations supporting the theoretical results.