Convergence of the Spectral Method for Stochastic Ginzburg-Landau Equation Driven by Space-Time White Noise

The main purpose of this paper is to investigate the spectral Galerkin method for spatial discretization. We combine it with the method introduced by Kloeden, Jentzen & Winkel in [12] for temporal discretization of stochastic partial differential equations and study pathwise convergence. We consider the case of colored noise, instead of the usual space-time white noise that was used before for the spatial discretization. The rate of convergence in uniform topology is estimated for the stochastic Burgers equation. Numerical examples illustrate the estimated convergence rate.

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“…In the following, for shorthand notation, all spatial integrals are over (0, 1) d . Therefore, by (14) and (15) …”

Section: Then There Exists a Stochastic Processmentioning

confidence: 98%

Full discretization of the stochastic Burgers equation with correlated noise

Blömker

Kamrani

Hosseini

2013

IMA Journal of Numerical Analysis

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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: 76%

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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Convergence of the Spectral Method for Stochastic Ginzburg-Landau Equation Driven by Space-Time White Noise

Cited by 42 publications

References 17 publications

Full discretization of the stochastic Burgers equation with correlated noise

Full discretization of the stochastic Burgers equation with correlated noise

Noise-induced oscillations in an actively mode-locked laser

The Numerical Approximation of Stochastic Partial Differential Equations

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