Combined Hermite   spectral-finite difference method   for the Fokker-Planck equation

Fok, Johnson C. M.; Guo, Bing; Tang, Tao

doi:10.1090/s0025-5718-01-01365-5

Cited by 61 publications

(48 citation statements)

References 49 publications

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“…The first two results, (3.12) and (3.13), can be obtained by the arguments similar to those given in [5,7].…”

Section: Approximation Properties Of Hermite Functionssupporting

confidence: 55%

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Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Ma¹,

Sun²,

Tang³

2005

SIAM J. Numer. Anal.

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Abstract. Hermite spectral methods are investigated for linear diffusion equations and nonlinear convection-diffusion equations in unbounded domains. When the solution domain is unbounded, the diffusion operator no longer has a compact resolvent, which makes the Hermite spectral methods unstable. To overcome this difficulty, a time-dependent scaling factor is employed in the Hermite expansions, which yields a positive bilinear form. As a consequence, stability and spectral convergence can be established for this approach. The present method plays a similar role in the stability of the similarity transformation technique proposed by Funaro and Kavian [Math. Comp., 57 (1991), pp. 597-619]. However, since coordinate transformations are not required, the present approach is more efficient and is easier to implement. In fact, with the time-dependent scaling the resulting discretization system is of the same form as that associated with the classical (straightforward but unstable) Hermite spectral method. Numerical experiments are carried out to support the theoretical stability and convergence results.

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“…The first two results, (3.12) and (3.13), can be obtained by the arguments similar to those given in [5,7].…”

Section: Approximation Properties Of Hermite Functionssupporting

confidence: 55%

“…Some of them are similar to those obtained in [5,6,7,19,23,24] and we will only briefly outline the proofs.…”

Section: Approximation Properties Of Hermite Functionsmentioning

confidence: 57%

Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Ma¹,

Sun²,

Tang³

2005

SIAM J. Numer. Anal.

Self Cite

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“…Numerical simulation of VPFP systems was studied by several authors, see e.g. [16,33,38,42,11]. However, up to our knowledge, the numerical resolution of the high field limit has still not been studied and in particular no numerical comparisons with solutions of (2.5) are available.…”

Section: 1mentioning

confidence: 99%

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Gosse¹,

Vauchelet²

2016

SIAM J. Sci. Comput.

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Abstract. Numerical resolution of two-stream kinetic models in strong aggregative setting is considered. To illustrate our approach, we consider an 1D kinetic model for chemotaxis in hydrodynamic scaling and the high field limit of the Vlasov-Poisson-Fokker-Planck system. A difficulty is that, in this aggregative setting, weak solutions of the limiting model blow up in finite time, therefore the scheme should be able to handle Dirac measures. It is overcome thanks to a careful discretization of the macroscopic velocity resulting of Vol'pert calculus: accordingly, a new well-balanced (WB) and asymptotic preserving (AP) numerical scheme is provided. Numerical simulations confirm a good behavior of solutions.

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“…In particular, the author shows that by choosing this parameter inversely proportional to the number of Hermite functions, only O(p) functions are needed in order to resolve p wavelengths in one spatial dimension. More recently, the question of the optimal choice of the scaling parameter has also been studied in the framework of spectral methods for the Fokker-Planck equation; see [23].…”

Section: (4b)mentioning

confidence: 99%

Spectral Methods for Multiscale Stochastic Differential Equations

Abdulle¹,

Pavliotis²,

Vaes³

2017

SIAM/ASA J. Uncertainty Quantification

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Abstract. This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.

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Combined Hermite spectral-finite difference method for the Fokker-Planck equation

Cited by 61 publications

References 49 publications

Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

Spectral Methods for Multiscale Stochastic Differential Equations

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