On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

Shen, Jie; Yu, Haijun

doi:10.1137/100795929

Cited by 12 publications

(8 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A spectral-Galerkin approximation was further introduced in [25] based on a weighted weak formulation for f (b − |m| 2 ) −b/4 . An improved weighted formulation was proposed in [36] in terms of f (b − |m| 2 ) −s/2 for 1 < s ≤ b, leading to a different spectral-Galerkin algorithm. We note that this weighted formulation was also used for specific values of s in [15,16,26] and was analyzed in section 3.2 of [25].…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

Liu¹,

Yu²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

In this paper, we propose an entropy satisfying conservative method to solve the Fokker-Planck equation of the finitely extensible nonlinear elastic dumbbell model for polymers, subject to homogeneous fluids. Both semidiscrete and fully discrete schemes satisfy all three desired properties-(i) mass conservation, (ii) positivity preserving, and (iii) entropy satisfying-in the sense that these schemes satisfy discrete entropy inequalities for both the physical entropy and the quadratic entropy. These ensure that the computed solution is a probability density and the schemes are entropy stable and preserve the equilibrium solutions. We also prove convergence of the numerical solution to the equilibrium solution as time becomes large. Zero flux at boundary is naturally incorporated, and boundary behavior is resolved sharply. Both one-and two-dimensional numerical results are provided to demonstrate the good qualities of the scheme and the effects of some canonical homogeneous flows.

show abstract

Section: Related Workmentioning

confidence: 99%

“…We note that this weighted formulation was also used for specific values of s in [15,16,26] and was analyzed in section 3.2 of [25]. The methods in [25,36] have provable stability results for certain weighted integrable initial data. However, positivity of the numerical solution is not guaranteed.…”

Section: Related Workmentioning

confidence: 99%

An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

Liu¹,

Yu²

2012

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…Thus, they provide a solid first step towards our ultimate goal of directly solving the coupled five-and six-dimensional Navier-Stokes Fokker-Planck equations. These results are presented in [5] and [6].…”

Section: Approximation Of the Fokker-planck Equation Of Fene Dumbbellmentioning

confidence: 99%

Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation

Shen¹,

Weinan²

2011

Self Cite

View full text Add to dashboard Cite

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Department of Defense, Washington Headquarters Services, Directorate for Information Approved for public release A major issue in modeling and computation is how to handle high dimen-sional problems. We can divide these high dimensional problems into two classes: moderately high dimensional problems or very high dimensional prob-lems. In the former class, we have problems such as the Boltzmann equation and Fokker-Planck equation, whose dimensionality is moderately high but are amendable to sparse grid based methods. In the latter class, we have problems such as exploration of the configuration space of a large molecule. These prob-lems often involve hundreds of thousands of dimensions, and methods based on fixed grids are far from being adequate. We developed various techniques in handling these problems using the hyperbolic cross/sparse representation for the former class, and adaptive sampling for the latter. These developments are aimed at providing a solid foundation for efficient and reliable numerical simulations of Boltzmann and Fokker-Planck equations. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S)Besides the work presented in this report, a number of other related publi-cations by the PIs were also partially supported by this grant. UU UU UUJie Shen 7654941923 solving boltzmann and fokker-planck equations using sparse representation AFOSR FA9550-08-1-0416Jie Shen Department of Mathematics, Purdue University Weinan E Department of Mathematics, Princeton University Abstract A major issue in modeling and computation is how to handle high dimensional problems. We can divide these high dimensional problems into two classes: moderately high dimensional problems or very high dimensional problems. In the former class, we have problems such as the Boltzmann equation and Fokker-Planck equation, whose dimensionality is moderately high but are amendable to sparse grid based methods. In the latter class, we have problems such as exploration of the configuration space of a large molecule. These problems often involve hundreds of thousands of dimensions, and methods based on fixed grids are far from being adequate. We developed various techniques in handling these problems using the hyperbolic cross/sparse representation for the former class, and adaptive sampling for the latter. These developments are aimed at providing a solid foundation for efficient and reliable numerical simulations of Boltzmann and Fokker-Planck equati...

show abstract

“…An alternative approach is the direct simulation of the equivalent Fokker-Planck equation in the configurational space. Examples include Galerkin spectral element technique 15,38,54,65,68 and the lattice Boltzmann technique. 1,7 As pointed out in Ref.…”

Section: Introductionmentioning

confidence: 99%

On a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids

Bao¹,

Li²,

Wang³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we propose a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids, which combines a finite element discretization to the macroscopic fluid dynamic equation with a variational particle scheme to the microscopic Fokker-Planck equation. The discretization is constructed by a discrete energetic variational approach, and preserves the microscopic variational structure in the semi-discrete level. Numerical examples demonstrate the accuracy and robustness of the proposed numerical scheme for some special external flows with a wide range of flow rates.

show abstract

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

Cited by 12 publications

References 26 publications

An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

An Entropy Satisfying Conservative Method for the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model

Solving Boltzmann and Fokker-Planck Equations Using Sparse Representation

On a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids

Contact Info

Product

Resources

About