On Numerical Approximation of Stochastic Burgers' Equation

Alabert, Aureli; Gyöngy, István

doi:10.1007/978-3-540-30788-4_1

Cited by 24 publications

(41 citation statements)

References 10 publications

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“…, we see that the BV-norm of G 1 ; 2 is bounded by 2 1 1 times the BV-norm of exp. 1 2 x 2 f .x//. This on the other hand is bounded uniformly in 1 by Assumption 1.2, so that we have established (6.42).…”

Section: Proofmentioning

confidence: 59%

“…We fix 0 Ä s 1 < s 2 Ä T and an 0 < r 1 < r 2 Ä 1. Then, using Hölder's inequality we get for any 0 < < 1 and any p > 0 that Eˇ.S " .r 1 / S " .r 2 //.X " .s 2 / X " .s 1…”

Section: The Stochastic Convolutionmentioning

confidence: 99%

“…R n (so that d t u D G.u/d x u would describe a system of conservation laws), and the general case. In the gradient case, existence and uniqueness for (1.1) has been known at least since the 1990s [7,17] and convergence results for numerical schemes have, for example, been obtained in [1,5].…”

Section: Introductionmentioning

confidence: 99%

“…This means that the chain rule is postulated and the nonlinearity is rewritten as G.u.t; x//@ x u.t; x/ D @ x G .u.t; x//; (1.3) which makes sense as a distribution as soon as u is continuous. The approximation schemes studied, for example, in [1,5] respect this conservation law structure by considering natural approximations of @ x G . For example, it is not difficult to show that if u " solves (1.4) @ t u " D @ 2…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Approximating Rough Stochastic PDEs

Hairer

Maas

Weber

2013

Comm Pure Appl Math

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We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed recently in Probability Theory Related Fields by Hairer and Weber. The key idea was to use the theory of controlled rough paths to give definitions of weak/mild solutions and to set up a Picard iteration argument. In this article the limiting behavior of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the Itô‐Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.© 2014 Wiley Periodicals, Inc.

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Section: Proofmentioning

confidence: 59%

“…We fix 0 Ä s 1 < s 2 Ä T and an 0 < r 1 < r 2 Ä 1. Then, using Hölder's inequality we get for any 0 < < 1 and any p > 0 that Eˇ.S " .r 1 / S " .r 2 //.X " .s 2 / X " .s 1…”

Section: The Stochastic Convolutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximating Rough Stochastic PDEs

Hairer

Maas

Weber

2013

Comm Pure Appl Math

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“…In recent years, some of the main numerical methods for solving stochastic partial differential equations (SPDEs), like finite difference and finite element schemes, have been considered [1]- [9], e.g. [10]- [12], based on a finite difference scheme in both space and time. It is well known that explicit time discretization via standard methods (e.g., as the Euler-Maruyama method) leads to a time step restriction due to the stiffness originating from the discretization of the diffusion operator (e.g.…”

Section: Introductionmentioning

confidence: 99%

Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs

Mohammed¹,

Sohaly²,

El-Bassiouny³

et al. 2014

AJCM

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Stochastic partial differential equations (SPDEs) describe the dynamics of stochastic processes depending on space-time continuum. These equations have been widely used to model many applications in engineering and mathematical sciences. In this paper we use three finite difference schemes in order to approximate the solution of stochastic parabolic partial differential equations. The conditions of the mean square convergence of the numerical solution are studied. Some case studies are discussed.

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Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

Arab

Debbi

2020

Math Methods in App Sciences

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In this work, we use the spectral Galerkin method to prove the existence of a pathwise unique mild solution of a fractional stochastic partial differential equation of Burgers type in a Hölder space. We get the temporal regularity, and using a combination of Galerkin and exponential-Euler methods, we obtain a full discretization scheme of the solution. Moreover, we calculate the rates of convergence for both approximations (Galerkin and full discretization) with respect to time and to space.

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On Numerical Approximation of Stochastic Burgers' Equation

Cited by 24 publications

References 10 publications

Approximating Rough Stochastic PDEs

Approximating Rough Stochastic PDEs

Mean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs

Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

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