Stochastic exponential integrators for a finite element discretisation of SPDEs with additive noise. (English) Zbl 1407.65198
Summary: We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square \(L^2\) norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35K10 | Second-order parabolic equations |
| 35K58 | Semilinear parabolic equations |
Keywords:
parabolic stochastic partial differential equations; finite element method; exponential integrators; higher-order approximation; strong numerical approximation; additive noise; transport in porous mediaSoftware:
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