Numerical simulation of stochastic PDEs for excitable media

Abstract. We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of the noise and the approximation of the exponential function by a rational fraction, we introduce a new scheme, designed for finite elements, finite volumes or finite differences space discretization, similar to the schemes in [4,7] for spectral methods and [1] for finite element methods. We use the projection operator, the smoothing effect of the positive definite selfadjoint operator and linear functionals of the noise in Fourier space to obtain higher order approximations. We consider noise that is white in time and either in H 1 or H 2 in space and give convergence proofs in the mean square L 2 norm for a diffusion reaction equation and in mean square H 1 norm in the presence of an advection term. For the exponential integrator we rely on computing the exponential of a non-diagonal matrix. In our numerical results we use two different efficient techniques: the real fast Léja points and Krylov subspace techniques. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.

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“…Consider the covariance operator Q with the following covariance function (kernel) with strong exponential decay [28,29] …”

Section: Real Fast Léja Points Techniquementioning

confidence: 99%

Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

Lord

Tambue²

2012

IMA Journal of Numerical Analysis

149

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“…The techniques explored here focus on methods that provide insight into the time-dependent aspects of the system but are more appropriate for the limited computational resources available in a grid-based environment. The techniques are similar to those proposed by Talay [10], Printems [11,12], Shardlow [13], and Liu [14].…”

Section: Introductionmentioning

confidence: 74%

“…This circulating pulse is very sensitive however to the presence of noise, as we will demonstrate below using a simple argument based on the phase plane. Our analysis here is motivated by and similar to that of Shardlow [13].…”

Section: Phase Plane Analysis Of the Role Of Noisementioning

confidence: 95%

“…The use of spectral approximations is the most relevant to the methods examined here. In particular, the approximation of the Barkley model for excitable media with additive noise was examined by Shardlow [13] with spectral expansions of the noise terms. A similar approach was examined by Liu [14] for the approximation of the Ginzburg-Landau equation.…”

Section: Numerical Approximation Of Temporal Noisementioning

confidence: 99%

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Noise-induced oscillations in an actively mode-locked laser

Black

Geddes

2010

Computers & Mathematics with Applications

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a b s t r a c tOscillations induced by noise are examined for an actively mode-locked laser. Additive noise, proportional noise, and combined noise are considered. Spatial noise is approximated by Hermite expansions and temporal noise is approximated via an approximation of the variance of the random variable using a fourth-order Adams-Bashforth scheme. The approach is verified on a sample problem and used to explore the governing equations for a mode-locked laser. The inclusion of multiplicative noise leads to much wider pulses and much longer intervals between pulses.

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“…Such methods were used in [4,1,2,18,22,23,27,35,37,31,42,43,45,46,49,50,53,54,79,71,73,78,75,83,84,85,95,96,106,103,107,112,113,114,115,108,121,122,123,125,127,128,129,130]. Beside this prevalent strategy also the splitting up approach for temporal discretizations (see [8,60,38,39,40,89]), the Wiener chaos expansion (see [57,…”

Section: Other Resultsmentioning

confidence: 99%

The Numerical Approximation of Stochastic Partial Differential Equations

2009

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The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered. Mathematics Subject Classification (2000). Primary 60H35; Secondary 60H15.

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Numerical simulation of stochastic PDEs for excitable media

Cited by 60 publications

References 17 publications

Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

Stochastic exponential integrators for the finite element discretization of SPDEs for multiplicative and additive noise

Noise-induced oscillations in an actively mode-locked laser

The Numerical Approximation of Stochastic Partial Differential Equations

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