Stability and convergence analysis of a fully discrete semi-implicit scheme for stochastic Allen-Cahn equations with multiplicative noise

Abstract

We consider a fully discrete scheme for stochastic Allen-Cahn equation in a multi-dimensional setting. Our method uses a polynomial based spectral method in space, so it does not require the elliptic operator A A and the covariance operator Q Q of noise in the equation commute, and thus successfully alleviates a restriction of Fourier spectral method for stochastic partial differential equations pointed out by Jentzen, Kloeden and Winkel [Ann. Appl. Probab. 21 (2011), pp. 908–950]. The discretization in time is a tamed semi-implicit scheme which treats the nonlinear term explicitly while being unconditionally stable. Under regular assumptions which are usually made for SPDEs, we establish strong convergence rates in the one spatial dimension for our fully discrete scheme. We also present numerical experiments which are consistent with our theoretical results.