Stochastic Partial Differential Equations : Spatially Adaptive Numerical Methods
Stochastic partial differential equations (SPDEs) pose significant mathematical and computational challenges not present in the corresponding deterministic PDE setting. Solutions to SPDEs are often not classical requiring instead a measure on a space of generalized functions (distributions). These features pose significant challenges for the numerical approximation of SPDEs. Often spectral methods are employed but this usually requires periodic boundary conditions or domains of rather simple geometry. We are developing alternative finite difference numerical methods for the approximation of SPDEs utilizing ideas from statistical mechanics. Our approach allows for general boundary conditions, complex domain geometries, and the use of adaptive spatial meshes. A key challenge these methods address for adaptive meshes is to properly treat the stochastic driving terms at the coarse-refined interfaces.
As a demonstration we present results for a stochastic reaction-diffusion system with two chemical species undergoing Gray-Scott reactions subject to intrinsic concentration fluctuations. The results show the evolution of a stochastically induced spatial pattern which does not occur in the absence of fluctuations. Our methods utilize a quad-tree adaptive refinement mesh which dynamically changes as corresponding spatial regions become chemically active.
- Growth of a noise-induced spatial pattern resolved with adaptive meshing [AVI Movie].
An important feature of the stochastic numerical methods is how the the stochastic driving field is treated at the coarse-refined interface. For a purely diffusive system the equilibrium fluctuations in concentration are essentially spatially uncorrelated at any instant. At coarse-refined interfaces the discretization of the stochastic driving field may introduce artificial long-range spatial correlations. Below is shown the resulting equilibrium covariance structure of fluctuations with the mesh site marked +. Results are shown for the cases corresponding to a white-noise discretized stochastic driving field, a discretized stochastic driving field derived from random fluxes, and our discretized stochastic driving field. To derive consistent discretizations of the stochastic driving field at coarse-refined interfaces, we use notions related to the fluctuation-dissipation principle of statistical mechanics. We are using these approaches to develop a variety of stochastic numerical methods for the study of spatially extended stochastic systems.
Select Publications
- Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (Journal of Computational Physics, Volume 229, Issue 9, 1 May, Pages 3474-3501, (2010). [PDF] [DOI]
- Spatially Adaptive Stochastic Multigrid Methods for Fluid-Structure Systems with Thermal Fluctuations, Atzberger, P.J., (preprint), (2010). [PDF]
Movies for Select Simulation Results
- Fluctuation-induced spatial pattern resolved with adaptive meshing: [AVI Movie].
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