Passive scalar equation in a turbulent incompressible Gaussian velocity field

We elucidate the effect of noise on the dynamics of N point charges in a Vlasov-Poisson model driven by a singular bounded interaction force. A too simple noise does not impact the structure inherited from the deterministic case and, in particular, cannot prevent coalescence. Inspired by the theory of random transport in passive scalars, we identify a class of random fields which generate random pulses that are chaotic enough to disorganize the deterministic structure and prevent any collapse of the particles. We obtain strong unique solvability of the stochastic model for any initial configuration of different point charges. In the case where there are exactly two particles, we implement the "vanishing noise method" for determining the continuation of the deterministic model after collapse.

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“…Bounding the resolvent of the linear system (25)(26) in terms of the bounds for x 1,ε , v 1,ε and v 2,ε , the result easily follows.…”

Section: Key Lemmas By Integration By Partsmentioning

confidence: 85%

Noise Prevents Collapse of Vlasov‐Poisson Point Charges

Delarue

Flandoli

Vincenzi

2013

Comm Pure Appl Math

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“…A prominent example of the stochastic advection-diffusion equation (2.1)-(2.2) is a passive scalar equation, which is motivated by the study of the turbulent transport problem (see [7,16,18] and the references therein). Here we perform numerical tests on the two-dimensional (d = 2) passive scalar equation with periodic boundary conditions:…”

Section: Numerical Tests With Passive Scalar Equationmentioning

confidence: 99%

A Multistage Wiener Chaos Expansion Method for Stochastic Advection-Diffusion-Reaction Equations

Zhang¹,

Rozovskiĭ²,

Tretyakov³

et al. 2012

SIAM J. Sci. Comput.

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Using Wiener chaos expansion (WCE), we develop numerical algorithms for solving second-order linear parabolic stochastic partial differential equations (SPDEs). We propose a deterministic WCE-based algorithm for computing moments of the SPDE solutions without any use of the Monte Carlo technique. We also compare the proposed deterministic algorithm with two other numerical methods based on the Monte Carlo technique and demonstrate that the new method is more efficient for highly accurate solutions. Numerical tests verify that the scheme is of mean-square order) for diffusion and for diffusion-reaction SPDEs with constant or variable coefficients, where Δ is the time step, and N is the Wiener chaos order. Introduction.In this paper we develop a new numerical method, based on nonlinear filtering ideas and spectral expansions, for advection-diffusion-reaction equations perturbed by random fluctuations, which form a broad class of second-order linear parabolic stochastic differential equations (SPDEs). The standard approach to constructing SPDE solvers starts with a space discretization of an SPDE, for which spectral methods (see, e.g., [4,10,14]), finite element methods (see, e.g., [1,8,32]) or spatial finite differences (see, e.g., [1,11,30,33]) can be used. The result of such a space discretization is a large system of ordinary stochastic differential equations (SDEs) which requires time discretization to complete a numerical algorithm. In [5,6] an SPDE is first discretized in time and then a finite element or finite difference method can be applied to this semidiscretization. Other numerical approaches include those making use of splitting techniques [2,17,12], quantization [9], or an approach based on the averaging-over-characteristic formula [26,27]. In [22,19] numerical algorithms based on the Wiener chaos expansion (WCE) were introduced for solving the nonlinear filtering problem for hidden Markov models. Since then the WCE-based numerical methods have been successfully developed in a number of directions (see, e.g., [13,31]).In computing moments of SPDE solutions, the existing approaches to solving SPDEs are usually complemented by the Monte Carlo technique. Consequently, in these approaches numerical approximations of SPDE moments have two errors: numerical integration error and Monte Carlo (statistical) error. To reach a high accuracy, we have to run a very large number of independent simulations of the SPDE to reduce the Monte Carlo error. Instead, here we exploit WCE numerical methods to construct a deterministic algorithm for computing moments of the SPDE solutions without any

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“…Since then, stochastic partial differential equations and stochastic models of fluid dynamics have been the object of intense investigations which have generated several important results. We refer, for instance, to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] . Similar investigations for Non-Newtonian fluids have almost not been undertaken except in very few work; we refer, for instance, to [23][24][25] for some computational studies of stochastic models of polymeric fluids.…”

Section: Introductionmentioning

confidence: 99%