Multirevolution Integrators for Differential Equations with Fast Stochastic Oscillations

Laurent, Adrien; Vilmart, Gilles

doi:10.1137/19m1243075

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…Numerical results by Belaouar, de Bouard, and Debussche in [6] led the authors to conjecture that this wellposedness in fact holds for all p < 9, in agreement with the scaling of (1). This conjecture is supported by the further numerical calculations by Cohen and Dujardin [18] and Laurent and Vilmart [30]. If this is indeed the case, it represents a regularization by noise phenomenon for the equation.…”

Section: Introductionmentioning

confidence: 55%

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On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Stewart¹

2022

Preprint

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We study the wellposedness of the periodic nonlinear Schrödinger equation with white noise dispersion and a power nonlinearity given byWe develop Strichartz estimates for this equation, which we then use to prove almost sure global wellposedness of this equation with L 2 initial data for nonlinearities with exponent 1 < p ≤ 3. By generalizing the Fourier restriction spaces X s,b to the stochastic setting, we also prove that our solutions agree with the ones constructed by Chouk and Gubinelli (Commun. Part. Diff. Eq. 40(11):2047-2081 using rough path techniques. We also consider the quintic equation (p = 5), and show that it is analytically illposed in L 1 ω CtL 2 x .

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Section: Introductionmentioning

confidence: 55%

“…x ∩ H 1 x exist globally and satisfy the average decay estimate E u(t) L ∞ x (1 + t) −1/4 . There have also been several works considering numerical schemes for (1) [6,18,19,30,33].…”

Section: Introductionmentioning

confidence: 99%

On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

Stewart¹

2022

Preprint

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“…Moreover, it reduces to the constrained Euler scheme (1.7) in the limit ε Ñ 0. Let us mention that uniformly accurate and asymptoticpreserving integrators are used extensively in the context of multiscale problems (see, for instance, the recent works [6,8,9,22], as well as the references therein), though the problem we study and the techniques we use in the present paper are different.…”

Section: Chap Vi-vii])mentioning

confidence: 99%

A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics

Laurent¹

2021

Preprint

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In molecular dynamics, penalized overdamped Langevin dynamics are used to model the motion of a set of particles that follow constraints up to a parameter ε. The most used schemes for simulating these dynamics are the Euler integrator in R d and the constrained Euler integrator. Both have weak order one of accuracy, but work properly only in specific regimes depending on the size of the parameter ε. We propose in this paper a new consistent method with an accuracy independent of ε for solving penalized dynamics on a manifold of any dimension. Moreover, this method converges to the constrained Euler scheme when ε goes to zero. The numerical experiments confirm the theoretical findings, in the context of weak convergence and for the invariant measure, on a torus and on the orthogonal group in high dimension and high codimension.

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“…Cohen [4] proposed an extension to the Kubo oscillator by including a non-linear, skew-symmetric drift term, see also Laurent and Vilmart [16]. We extend this further, with non-linear terms in both the drift and diffusion, in addition to including multidimensional noise:…”

Section: Introductionmentioning

confidence: 97%

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

2022

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In this paper, we consider a class of stochastic midpoint and trapezoidal Lawson schemes for the numerical discretization of highly oscillatory stochastic differential equations. These Lawson schemes incorporate both the linear drift and diffusion terms in the exponential operator. We prove that the midpoint Lawson schemes preserve quadratic invariants and discuss this property as well for the trapezoidal Lawson scheme. Numerical experiments demonstrate that the integration error for highly oscillatory problems is smaller than that of some standard methods.

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Multirevolution Integrators for Differential Equations with Fast Stochastic Oscillations

Cited by 5 publications

References 25 publications

On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

On the wellposedness for periodic nonlinear Schrödinger equations with white noise dispersion

A uniformly accurate scheme for the numerical integration of penalized Langevin dynamics

Lawson schemes for highly oscillatory stochastic differential equations and conservation of invariants

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