J Sci Comput (2015) 62:803–830DOI 10.1007/s10915-014-9878-1

The Entropy Satisfying Discontinuous Galerkin Methodfor Fokker–Planck equations

Hailiang Liu · Hui Yu

Received: 3 February 2014 / Revised: 24 April 2014 / Accepted: 3 June 2014 / Published online: 18 June 2014© Springer Science+Business Media New York 2014

Abstract In Liu and Yu (SIAM J Numer Anal 50(3):1207–1239, 2012), we developed afinite volume method for Fokker–Planck equations with an application to finitely exten-sible nonlinear elastic dumbbell model for polymers subject to homogeneous fluids. Themethod preserves positivity and satisfies the discrete entropy inequalities, but has only firstorder accuracy in general cases. In this paper, we overcome this problem by developinguniformly accurate, entropy satisfying discontinuous Galerkin methods for solving Fokker–Planck equations. Both semidiscrete and fully discrete methods satisfy two desired properties:mass conservation and entropy satisfying in the sense that these schemes are shown to satisfythe discrete entropy inequality. These ensure that the schemes are entropy satisfying and pre-serve the equilibrium solutions. It is also proved the convergence of numerical solutions tothe equilibrium solution as time becomes large. At the finite time, a positive truncation is usedto generate the nonnegative numerical approximation which is as accurate as the obtainednumerical solution. Both one and two-dimensional numerical results are provided to demon-strate the good qualities of the schemes, as well as effects of some canonical homogeneousflows.

Keywords Fokker–Planck equations · FENE model · Entropy satisfying ·Discontinuous Galerkin

Mathematics Subject Classification 35K20 · 65M60 · 76A05 · 82C31 · 82D60

H. Liu (B)Mathematics Department, Iowa State University, Ames, IA 50011, USAe-mail: hliu@iastate.edu

H. YuInstitut de Mathmatiques de Toulouse, Toulouse, Francee-mail: hyu@math.univ-toulouse.fr

123

804 J Sci Comput (2015) 62:803–830

1 Introduction

In this paper, we are interested in constructing high order accurate schemes for solving theinitial boundary value problem for Fokker–Planck equations,

⎧⎨

⎩

∂t f = 12∇m · (∇m f − F f ), m ∈ B, t > 0,

∂ν f + F · ν f |∂ B = 0, t > 0,

f (m, 0) = f0(m), m ∈ B,

(1.1)

where f denotes the probability density function (pdf), F : B → Rd a given force field, m is

the configuration variable in a bounded domain B ⊂ Rd , and ν is the outward normal vector

on the boundary ∂ B.The Fokker–Planck equation is basic in many areas of physics. The main properties of

the solution to (1.1) are the nonnegativity principle, the mass conservation and the entropyinequality, i.e.,

f0 ≥ 0 �⇒ f ≥ 0 ∀t > 0, (1.2)∫

B

f (m, t) dm =∫

B

f0(m) dm ∀t > 0, (1.3)

and for any convex function H : R → R,

d

dt

∫

B

M H

(f

M

)

dm = −1

2

∫

B

M H ′′(

f

M

) ∣∣∣∣∇m

f

M

∣∣∣∣

2

dm ≤ 0, (1.4)

which can be shown to hold if M is an equilibrium solution to (1.1). These properties arealso naturally desired for numerical methods solving (1.1). The entropy satisfying methoddeveloped in [44] does satisfy all these three properties, but has only first order accuracy ingeneral cases. The maximum-principle-satisfying method recently developed in [45] is lim-ited to third order. In this paper, we develop uniformly accurate, entropy satisfying discontin-uous Galerkin methods for solving Fokker–Planck equations, coupled with a post-processingusing the positive truncation. By entropy satisfying we mean that the entropy relation (1.4) ismimicked by the proposed numerical method; we refer to [4,49] for the intrinic connectionsbetween the problem of trend to equilibrium for the Fokker–Planck equation of statisticalphysics and entropy inequalities.

1.1 Motivation Model

Our investigation is motivated by the study of the Fokker–Planck equation of the finitelyextensible nonlinear elastic (FENE) dumbbell model, with the force field defined by

F = 2κm − bm

b − |m|2 , m ∈ B := {|m|2 ≤ b}, (1.5)

which indicates that the FENE model takes into account the finite extensibility of the poly-mer chain, through an important explosive force as |m| tends to

√b, the maximum spring

extension. In this work, we assume κ is a real matrix with trace zero, which corresponds tohomogeneous incompressible fluids with the velocity field v = κx . Also we consider thecase of b > 2. Indeed, if b < 2, the FENE model yields many solutions unless a furtherrequirement on the boundary behavior of f is imposed; see [41,42]. A general discussion ofthis problem and background references are given in the introduction to [44].

123

J Sci Comput (2015) 62:803–830 805

Our interest is to develop higher order entropy satisfying methods for Fokker–Planckequations subject to various forces. Force field (1.5) is of particular interest since it involvestwo special difficulties: (i) the singularity of the Fokker–Planck equation near |m| = √

b,and (ii) nonconservative force due to the fluid coupling. These together present numerouschallenges, both analytically and numerically. The boundary singularity issue is particularlyimportant in the presence of fluid coupling, in which the behavior of the polymer distributionnear the boundary is of significance. Consequently, computing with sharp resolution andstability near the boundary is a major goal. In order to achieve this goal, three main propertiesof the solution of the Fokker–Planck equations are naturally desired for numerical schemessolving the equation: including the nonnegativity principle, the mass conservation and theexistence of nonzero steady state (equilibrium). For Fokker–Planck type equations, it is oftena challenge to find a proper formulation and an appropriate discretization to satisfy all thesenatural constraints; see the lecture notes [39, Section 4.3].

Our idea is to reformulate the Fokker–Planck equation (1.5a) as

∂t f = 1

2∇m ·

(

M∇m

(f

M

)

+ Fa f

)

, (1.6)

where M = e−U with Fa = F + ∇mU , with U chosen to resolve the singularity in theequation. With this formulation, we may use the relative entropy to test the stability of themodel as well as its numerical approximation. The entropy satisfying property will be testedusing H = g2 in (1.4), which is particularly convenient for higher order Galerkin methods.The usual physical entropy with H(g) = g log g is bounded by the quadratic entropy sinceMg log g ≤ Mg2 − Mg and the conservation of

∫Mg. In the coupled Navier–Stokes–

Fokker–Planck system both the entropic entropy [41] and the physical entropy [48] canensure the balance of total energy.

We should point out that for nonhomogeneous flows which is the case when consideringthe coupled system with the Navier-Stokes equation, one may apply the method developedin this paper using operator splitting techniques, see [15,16].

1.2 Discretization

As for the configurational discretization, we explore higher order discontinuous Galerkin(DG) methods based on formulation (1.6), while the finite volume method introduced in[44], satisfying all three desired properties, may be viewed as a first order DG method.The DG method is a finite element method using a completely discontinuous piecewisepolynomial space for the numerical solution and the test functions. One main advantage ofthe DG method was the flexibility afforded by local approximation spaces combined withthe suitable design of numerical fluxes crossing cell interfaces. In this work we follow themethodology of the Direct Discontinuous Galerkin (DDG) method. The main feature in theDDG schemes proposed in [40,43] for diffusion problems lies in numerical flux choices forthe solution gradient, which involves higher order derivatives evaluated crossing interfaces,motivated by a trace formula for the derivatives of the heat solution [40].

However, the DDG numerical flux given in [40,43] cannot be directly used, due to severalfeatures of (1.6): (1) the weight M has no positive lower bound, but zero at the boundary,making (1.6) singular and numerical computations more difficult; (2) the force due to fluideffects is generally nonconservative so that one has to consider a nonsymmetric perturbationupon the usual Landau formulation for Fokker–Planck equations, which makes the study oflong time convergence more interesting; (3) the natural function space for f is M L2(Mdm),

123

806 J Sci Comput (2015) 62:803–830

which when M vanishes at the boundary is different from the usual weighted space L2(Mdm)

[41,42].More precisely, this work involves several steps: we first apply the DG discretization to

(1.6) with f/M replaced by g, including interface corrections determined by a DDG typenumerical flux, in one dimensional case, of the form

∂m g = β0

h[g] + ∂m g + hβ1[∂2

m g];we then identify a key quantity �(β1) such that when β0 > 2�(β1)M , the resulting semidis-crete DG scheme satisfies the discrete version of entropy inequality (1.4) with H(g) = g2.We further present an approach to characterize � in terms of M , which shows that the aboveflux satisfying β0 > 2�(β1)M becomes adaptive in terms of both interface location and thesize of the elongation parameter b. This adaptive feature brings several advantages: (i) it hasthe ability of handling a wide range of b, particularly for more realistic case when b is large;(ii) when F is conservative, the method captures the long time behavior of solutions verywell, as predicted by the analysis and evidenced by the two dimensional tests on both thesimple extensional flow and the vortex flow. Indeed, for both semidiscrete and fully discreteDG schemes, we prove that numerical solutions converge towards the equilibrium state astime becomes large, i.e.,

limt→∞ f (m, t) = C M(m) where C =

∫

B f0 dm∫

B M dm.

We also provide an approach to construct a nonnegative numerical approximation from theobtained numerical solution at final time, without destroying the accuracy of the numericalsolution.

For F given in (1.5) with general κ such as the shear flow, the method is entropy stablein any finite time; see Theorem 4.4 in Sect. 3. Indeed the numerical results show quite goodconvergence for short time simulations; see Example 5 in Sect. 4.

1.3 Related Work

For the force field (1.5), the regime under consideration is b > 2, for which the zero fluxboundary condition is shown to be equivalent to the sharp requirement for the solution toremain a probability density [42]:

f = o(b − |m|2) on ∂ B. (1.7)

For theoretical results concerning the existence of solutions of the coupled system we refer to[12,42,47,48,60]; see also the works [18–20], and the earlier works on this problem [34,35].For rigorous analysis of long-time asymptotics of the FENE model, see [3,33]; and [4] forentropy methods to study rate of convergence to the equilibrium for Fokker–Planck typeequations. For some special configuration solutions with small flow rates, the use of momentclosure approximations has been investigated by several authors; see e.g. [27,30,31,53,57].Most numerical methods developed for the Fokker–Planck equation have been based on theform of (1.1); see, for example, [1,2,28,58]. Some elaborate numerical algorithms based onspectral methods were recently developed for the Fokker–Planck equation of the FENE modelin [15,16,46]. A spectral Galerkin approximation was further introduced in [36] based on

a weighted weak formulation for f (m, t)(b − |m|2)− b4 . An improved weighted formulation

was proposed in [55] in terms of f (m, t)(b − |m|2)−s for 1 < s ≤ b2 , leading to a different

spectral-Galerkin algorithm. We note that this weighted formulation was also used for specific

123

J Sci Comput (2015) 62:803–830 807

values of s in [15,16,46], and was analyzed in Sect. 2.2 of [36]. The methods in [36,55] haveprovable stability results for certain weighted integrable initial data.

The application of DG methods to hyperbolic problems has been quite successful, e.g.,Reed and Hill [51] for solving linear equations, and Cockburn et al. [22,24,25] for solvingnonlinear equations. However, the application of the DG method to diffusion problems hasbeen a challenging task because of the subtle difficulty in defining an appropriate numericalflux for the solution gradient; see the earlier works [5,7,59] using the interior penalty (IP)method. In past decade there has been a renewed interest in developing DG methods to solvethe diffusion problem, including the method originally proposed by Bassi and Rebay [8]for compressible Navier-Stokes equations, its generalization called the local discontinuousGalerkin (LDG) methods introduced in [26] and further studied in [13,21,23], as well asthe method introduced by Baumann-Oden [9,50]. We refer to [6] for the unified analysis ofDG methods for elliptic problems and background references for the IP methods. The directdiscontinuous Galerkin methods introduced in [40,43] adopt a different strategy, which is tosolve the higher order PDE directly by the DG discretization with the special numerical fluxfor the solution gradient, yet without rewriting the equation into a first order system. Thereare other recent works sharing the direct feature, such as those by Van Leer and Nomura in[56], Gassner et al. in [29], and Cheng and Shu in [17]. More general information about DGmethods for elliptic, parabolic and hyperbolic partial differential equations can be found inthe recent books and lectures notes [32,37,52,54].

Finally we comment on the concept of entropy explored in numerical approximations.There is a vast literature on entropic schemes for related equations including hyperbolicconservation laws and kinetic equations such as Fokker–Planck type equations. For Fokker–Planck equations subject to conservative forces such as F = −∇U with M = e−U , oneoften uses the logarithmic Landau form,

∂t f = 1

2∇ ·

(

f ∇ logf

M

)

,

to ensure the entropy property at discrete levels; see e.g., [11]. Another class of finite dif-ference schemes for Fokker–Planck equations is due to Chang-Cooper; see [14,38]. Thismethod is based upon the requirement that the discrete Fokker–Planck operator possesses aquasi-equilibrium solution which agrees at the mesh points with a quasi-equilibrium solutionof the analytic operator. For a linear Fokker–Planck equation the Chang-Cooper scheme isshown in [10] to make the underlying equation equivalent to the nonlogarithmic Landau form

∂t f = 1

2∇ ·

(

M∇ f

M

)

at the discrete level.

1.4 Contents

This paper develops high order discontinuous Galerkin methods for problem (1.5). InSect. 2, we describe our scheme for the one dimensional case. Theoretical analysis for bothsemidiscrete and fully discrete schemes is provided. In Sect. 3, we generalize the scheme totwo spatial dimensions. Numerical results of both one and two dimensions are presented inSect. 4. Finally, in Sect. 5, concluding remarks are given.

123

808 J Sci Comput (2015) 62:803–830

2 One-Dimensional Fokker–Planck Equation

In the one dimensional case, F is always conservative and can be written as F = −∇mU ,the Fokker–Planck problem for (1.1) becomes

2M∂t g = ∂m(M∂m g) in B × (0,∞),

g(m, 0) = g0(m) in B,

M∂m g = 0 on ∂ B × (0,∞),

(2.1)

where B = [−√b,

√b] and M(m) reduces to

M(m) = e−U

which is (b − m2)b2 for force field (1.5).

The relative entropy relation then reads

d

dt

∫

B

Mg2dm +∫

B

M |∂m g|2dm = 0. (2.2)

We shall construct a discontinuous Galerkin ( DG ) method to compute the numerical solutiong to (2.1), and then obtain f by using f = Mg, such that the entropy dissipation relation(2.2) is satisfied at the discrete setting.

2.1 Semidiscrete DG

We first discretize the equation with respect to m. Let

−√b = m 1

2< m 3

2< · · · < m N− 1

2< m N+ 1

2= √

b

be a partition of the domain B into subintervals I j = [m j− 12, m j+ 1

2] of length h j = m j+ 1

2−

m j− 12, j = 1, . . . , N . The center of I j is defined as

m j = m j− 12

+ 1

2h j .

We use the standard notation

[g] = g+ − g− and g = g+ + g−

2,

where g− and g+ are the values of g at m j+ 12

from the left and right cells respectively. Letthe solution space Vh be defined as

Vh = {v : v|I j ∈ Pk(I j ), j = 1, . . . , N },where Pk(I j ) is the space of polynomials of degree up to k on I j . We propose the followingDG method: find gh ∈ Vh such that for 1 ≤ j ≤ N ,

2∫

I j

M∂t ghv dm = −∫

I j

M∂m gh∂mv dm + M[∂m ghv + (gh − gh)∂mv]∣∣∣m

j+ 12

mj− 1

2

∀v ∈ Vh,

(2.3)

123

J Sci Comput (2015) 62:803–830 809

For simplicity of analysis, we formulate the scheme with a uniform mesh size h. Followingthe DDG flux in [43], we take

∂m gh = β0

h[gh] + ∂m gh + β1h[∂2

m gh] and gh = gh, (2.4)

where the parameters (β0, β1) may vary with j . The boundary contribution at m 12

and m N+ 12

are taken to be zero to incorporate the zero flux condition. The initial data gh(m, 0) isgenerated by the weighted L2 projection

∫

I j

M(gh(m, 0) − g0(m))v(m) dm = 0 ∀v ∈ Vh . (2.5)

The global DG formulation is obtained by summing (2.3) over all cells

2N∑

j=1

∫

I j

M∂t ghv dm + A(gh, v) = 0 ∀v ∈ Vh, (2.6)

where the bilinear operator A is defined as

A(gh, v) =N∑

j=1

∫

I j

M∂m gh∂mv dm +N−1∑

j=1

M(∂m gh[v] + [gh]∂mv)

∣∣∣m

j+ 12

.

The algorithm is well defined once the parameters β0 and β1 are chosen.

2.2 Numerical Flux and Coercivity of A

The form (2.4) makes the numerical flux adopted in (2.3) both consistent and conservative.We proceed to discuss how to choose the parameters (β0, β1) to ensure the coercivity of A,and hence the entropy satisfying property.

To this end, we define the discrete energy norm (it is a semi-norm) of v ∈ Vh

‖v‖2E =

N∑

j=1

∫

I j

M |∂mv|2dm +N−1∑

j=1

β0

hM[v]2

∣∣∣m

j+ 12

(2.7)

and the quantity

� j := �(β1, w j ) = supu∈Pk−1([−1,1])

(u(1) − 2β1∂x u(1))2

∫ 1−1 w j (x)u2(x)dx

. (2.8)

Here the weight functions w j (x) are defined as

w j (x) = min{w jl(x), w jr (x)} for 1 ≤ j ≤ N − 1,

with

w jl(x) = M

(

m j + h

2x

)

and w jr (x) = M

(

m j+1 − h

2x

)

on [−1, 1].

It is clear that � j depends on β1, the weight w j , and the polynomial degree k.The main result in this section is the following.

123

810 J Sci Comput (2015) 62:803–830

Theorem 2.1 (Coercivity) If on each cell interface m j+ 12

for 1 ≤ j ≤ N − 1,

β0 > 2� j M j+ 12

> 0, (2.9)

where M j+ 12

= M(m j+ 12), then there exists γ ∈ (0, 1) such that

A(v, v) ≥ γ ‖v‖2E ∀v ∈ Vh . (2.10)

To prove this result, we need the following lemma.

Lemma 2.2 For any v ∈ Vh,

(2∂mv + β1h[∂2mv])2

∣∣m

j+ 12

≤ 4� j

h

⎛

⎜⎝

∫

I j

+∫

I j+1

⎞

⎟⎠ M |∂mv|2 dm. (2.11)

Proof For any v ∈ Vh , we have u ∈ Pk−1([−1, 1]) if u is defined as

u(x) = ul(x) + ur (x) := ∂mv(m j + h

2x) + ∂mv(m j+1 − h

2x),∀x ∈ [−1, 1].

This gives

(2∂mv + β1h[∂2mv])2

∣∣m

j+ 12

= (u(1) − 2β1∂x u(1))2

≤ � j

1∫

−1

w j (x)u2(x)dx

≤ 2� j

⎛

⎝

1∫

−1

w jl(x)u2l (x) dx +

1∫

−1

w jr (x)u2r (x) dx

⎞

⎠ ,

which yields (2.11). ��

Using this key estimate we are able to establish the coercivity of the bilinear operator A.

Proof of Theorem 2.1 Using the Young inequality, we have

A(v, v) =N∑

j=1

∫

I j

M |∂mv|2 dm +N−1∑

j=1

M[v](

β0

h[v] + 2∂mv + β1h[∂2

mv])∣∣∣∣m

j+ 12

≥N∑

j=1

∫

I j

M |∂mv|2 dm +N−1∑

j=1

M

[β0 − α

h[v]2 − h

4α

(2∂mv + β1h[∂2

mv])2]∣∣∣∣m

j+ 12

,

(2.12)

for any 0 < α < β0. Take γ ∈ (0, 1) such that

β0 = 2

(1 − γ )2 � j M j+ 12,

123

J Sci Comput (2015) 62:803–830 811

and α = (1 − γ )β0 so that β0 − α = γβ0. Then, in virtue of (2.11), we have

A(v, v) ≥N∑

j=1

∫

I j

M |∂mv|2 dm+γ

N−1∑

j=1

β0

hM[v]2

∣∣∣m

j+12

− 1−γ

2

N−1∑

j=1

⎛

⎜⎝

∫

I j

+∫

I j+1

⎞

⎟⎠ M |∂mv|2 dm

≥γ

N∑

j=1

∫

I j

M |∂mv|2 dm + γ

N−1∑

j=1

β0

hM[v]2

∣∣∣m

j+ 12

=γ ‖v‖2E .

��Due to the special role played by � j in the determination of β0, we proceed to give some

refined characterization of � j .

Lemma 2.3 It holds� j = ρ(H− 1

2 O H− 12 ), (2.13)

where ρ(·) denotes the spectral radius of a matrix and the two matrices O = (Oμν) andH = (Hμν) are defined by

Oμν = [1 − 2β1(μ − 1)][1 − 2β1(ν − 1)], 1 ≤ μ, ν ≤ k,

and

Hμν =1∫

−1

w(x)φμ(x)φν(x) dx =1∫

−1

w(x)xμ+ν−2 dx, 1 ≤ μ, ν ≤ k.

Proof If the basis {φμ}kμ=1 of Pk−1([−1, 1]) is taken as {xμ−1}k

μ=1, then u can be represented

as u(x) = ∑kμ=1 aμxμ−1 = �aT φ(x). Then

(u(1) − 2β1∂x u(1))2 = �aT O �a and u2(x) = �aT φ(x)φT (x)�a.

�aT H �a =1∫

−1

w j (x)u(x)2 dx > 0 ∀�a �= 0,

impliess that H is both symmetric and positive definite. So is H± 12 . Let y = H

12 �a, and we

have

�aT O �a = yT H− 12 O H− 1

2 y ≤ ρ(H− 12 O H− 1

2 )|y|2 = ρ(H− 12 O H− 1

2 )�aT H �a.

This yields (2.13) as desired. ��2.3 Entropy Stability

Our DG scheme (2.6) then has the following properties.

Theorem 2.4 The semidiscrete DDG (2.6) with (2.9) satisfies the following properties:

(1) Conservation of mass:N∑

j=1

∫

I jfh(m, t) dm = ∫

B f0(m) dm ∀t ≥ 0.

123

812 J Sci Comput (2015) 62:803–830

(2) The semidiscrete relative entropy

E(t) =N∑

j=1

∫

I j

Mg2h(m, t) dm

is nonincreasing in time. More precisely, there exists γ ∈ (0, 1) such that

d

dtE(t) ≤ −γ ‖gh‖2

E ≤ 0. (2.14)

Moreover,

E(t) ≤ E(0) ≤∫

B

Mg20(m) dm ∀t ≥ 0.

Proof (1) Conservation of mass follows from taking v = 1 in (2.6).(2) (2.6) with v = gh gives

d

dtE(t) = 2

∫

B

M∂t gh gh dm = −A(gh, gh),

which when using the coercivity estimate in Theorem 2.1 yields (2.14). Then

E(t) ≤ E(0) =N∑

j=1

∫

I j

Mg2h(m, 0) dm ≤

∫

B

Mg20(m) dm,

where we have used the projection (2.5). ��We may also examine the large time behavior of gh .

Theorem 2.5 Consider the semidiscrete scheme (2.6). The numerical solution fh(m, t) =gh(m, t)M(m) converges to C M(m), as t → ∞, where

C =∫

B f0(m) dm∫

B M(m) dm. (2.15)

Proof We introduce a set of functions by

=

⎧⎪⎨

⎪⎩w ∈ V k

h ;N∑

j=1

∫

I j

Mw dm =∫

B

f0(m) dm

⎫⎪⎬

⎪⎭.

On this set we define a functional V [gh] by

V [gh] =N∑

j=1

∫

I j

(gh − C)2 M dm.

Let g(x) ≡ C . We see that g ∈ and ddt g = 0.

A direct verification shows that

V [gh] = E(t) − C∫

B

f0(m) dm,

which implies that ddt V = d

dt E . Hence V satisfies the following:

123

J Sci Comput (2015) 62:803–830 813

• V [gh] > 0 for any gh ∈ \ {g}(positive definite),• d

dt V [gh] ≤ 0 for all gh ∈ (negative semidefinite),• the set { d

dt V (gh) = 0}⋂ does not contain any trajectories of the ODE system (2.6)besides the trajectory gh(m, t) = g, ∀t > 0.

With these properties we can apply the Krasovskii-LaSalle principle to conclude that

limt→∞ gh(m, t) = g,

which leads to the conclusion. It is left to verify the stated properties of V . The first twoproperties of V are easy to verify. We only verify the third property: from (2) of Theorem2.4 it follows that if d

dt E(t) = 0, then

‖gh‖2E =

N∑

j=1

∫

I j

M |∂m gh |2dm +N−1∑

j=1

β0

hM[gh]2

∣∣∣

j+ 12

= 0.

From the first summation, we have ∂m gh = 0 on all computational cells. Therefore gh mustbe a constant on each cell. The second summation gives that [gh] = 0 on all the cell interfaces.They both ensure that gh = const on B. While within , gh = g must hold. The proof isthus complete. ��2.4 Fully Discrete DG

Let gnh (m) denote the numerical solution gh at t = tn and �t the time step. We take the

following fully discrete DG scheme:

2N∑

j=1

∫

I j

Mgn+1

h − gnh

�tvdm = −A(g∗

h , v) ∀v ∈ Vh . (2.16)

Here g∗ is defined as

g∗h = ξgn+1

h + (1 − ξ)gnh ,

1

2≤ ξ ≤ 1.

Theorem 2.6 The fully discrete scheme (2.16) has a unique solution {gnh }. Moreover, the

solution satisfies the following properties:

(1) Conservation of mass:N∑

j=1

∫

I j

Mgn+1h dm = ∫

BMg0 dm.

(2) The relative entropy

En =N∑

j=1

∫

I j

M(gnh )2 dm

satisfies the following inequality:

En+1 ≤ En − �tγ ‖g∗h‖2

E − (2ξ − 1)

N∑

j=1

∫

I j

M(gn+1h − gn

h )2 dm ≤ En . (2.17)

123

814 J Sci Comput (2015) 62:803–830

Proof We only need to show the uniqueness of the solution for a finite dimensional problem.Assume there are two solutions to the linear system, gn+1

h1 and gn+1h2 . Then the difference

e = gn+1h1 − gn+1

h2 satisfies

2N∑

j=1

∫

I j

Mev dm = −�t A(ξe, v).

This with v = e gives

2N∑

j=1

∫

I j

Me2 dm = −�tξ A(e, e).

This holds true only if e = 0, which gives the uniqueness as well as the existence of thenumerical solution.

(1) (2.16) with v = 1 gives the conservation of mass.(2) Using v = g∗

h and

2g∗h = (gn+1

h + gnh ) + (2ξ − 1)(gn+1

h − gnh )

in (2.16), we obtain

En+1 − En = −�t A(g∗h , g∗

h) − (2ξ − 1)

N∑

j=1

∫

I j

M(gn+1h − gn

h )2 dm. (2.18)

Hence with ξ ∈ [ 12 , 1] and the coercivity of A, we have

En+1 ≤ En .

��Moreover, we have

Theorem 2.7 Let gnh be the numerical solution to the fully discrete DG scheme (2.16) with

12 < ξ ≤ 1, then it converges as n → ∞ with

gnh → C,

where C is defined in (2.15).

Proof Since En is nonincreasing and bounded from below, we have

limn→∞ En = inf{En}.

Observe from (2.18) that En+1 − En is a sum of nonpositive and bounded terms. Whenpassing the limit n → ∞ we conclude that each term must have zero as its limit, that is

limn→∞

N∑

j=1

∫

I j

M |∂m g∗|2dm = 0, limn→∞

N−1∑

j=1

β0

hM[g∗]2

∣∣∣

j+ 12

= 0

and limn→∞

N∑

j=1

∫

I j

M(gn+1 − gn)2 dm = 0 (2.19)

123

J Sci Comput (2015) 62:803–830 815

The first and third relations in (2.19) tell that gnh has limit and the limit must be constant in

each computational cell. The second relation in (2.19) infers that the limit must be C overthe whole domain B. The proof is complete. ��2.5 A Positive Approximation

Consider a positive truncation operator

u → u+ = max(u, 0) = 1

2(|u| + u)

for any integrable function defined on a Lipschitz domain . It is a well-known result thatthis operator defines a contraction in L p( ), which is bounded in W 1,p( ), ∀p ∈ [1,+∞],that is

‖u+ − v+‖L p( ) ≤ ‖u − v‖L p( ), (2.20)

‖u+‖W 1,p( ) ≤ ‖u‖W 1,p( ). (2.21)

Hence we construct a nonnegative approximation from the obtained numerical solution atthe final time by

gn |I j = (gnh )+, m ∈ I j , j = 1, . . . , N . (2.22)

We then have the following statement.

Theorem 2.8 The nonnegative piecewise polynomial gn defined by (2.22) is as accurate asthe original polynomial gn

h in any L p norm (1 ≤ p ≤ ∞). Moreover,

‖gnh − w‖L p ≤ ‖gn

h − w‖L p .

for any nonnegative function w.

Proof A direct calculation using the contraction property (2.20) gives

‖gnh − w‖p

L p =N∑

j=1

‖gnh − w‖p

L p(I j )

=N∑

j=1

‖(gnh )+ − w+‖p

L p(I j )

≤N∑

j=1

‖gnh − w‖p

L p(I j )= ‖gn

h − w‖pL p .

��Remark 2.1 The above truncation trick can be applied to a wider class of problems than thepresent one. In particular, the contraction property should be helpful in the control of the erroraccumulation if such a truncation needs to be implemented at every time step. However, inthis paper, the construction at immediate time steps is not used since it is numerically shownto destroy the mass conservation, hence producing inaccurate solutions.

Finally we come to a point to show how fully discrete solutions gnh are represented: if we

take the basis function {φl}N (k+1)l=1 , then

gnh =

(k+1)N∑

l=1

anl φl ,

123

816 J Sci Comput (2015) 62:803–830

where {φl} j (k+1)

l=( j−1)(k+1)+1 is the basis of Pk on the cell I j and zero in any other cell. Problem

(2.16) is then equivalent to a linear system with an unknown vector �an = (an1 , . . . , an

N (k+1))T

satisfying

(W − �tξ F)�an+1 = (W + �t (1 − ξ)F)�an .

where

W = 2N∑

j=1

∫

I j

MφφT dm, F = −A(φT , φ).

3 Extension to the Multidimensional FENE Model

In this section, we focus on the two dimensional force field given in (1.5) with T r(κ) = 0.The methods and related results can be extended to 3D case using the spherical coordinates.Set

M(m) = (b − |m|2) b2 exp(mT κsm), (3.1)

where κs is the symmetric part of κ , and κa = κ − κs is the antisymmetric part. TheFokker–Planck equation can be rewritten as

∂t f = 1

2∇m ·

(

M∇m

(f

M

)

− 2κam f

)

. (3.2)

In terms of g = fM , (3.2) becomes

2M∂t g = ∇m · [M∇m g − 2Mκamg]. (3.3)

In 2D case, κ has the following form,

κ =(

κ11 κ12

κ21 −κ11

)

,

with

κs =(

κ11κ12+κ21

2κ12+κ21

2 −κ11

)

,

and

κa =(

0 κ12−κ212− κ12−κ21

2 0

)

= a

(0 1

−1 0

)

with a = κ12 − κ21

2.

If κ is normal, i.e. κκT = κT κ , then ∇m · (Mκam) = 0 [see (3.14)]; M is hence theequilibrium solution to (3.2). For general κ , M is not necessarily the equilibrium solution,yet we still use the functional

E(t) =∫

B

Mg2 dm,

to measure the stability of the method. Our goal is to design a high order numerical methodwhich satisfies the discrete entropy inequality for κ normal, and entropy stable in the senseof E(t) ≤ C E(0) for general κ .

123

J Sci Comput (2015) 62:803–830 817

3.1 Discretization

Due to the geometry of the domain B, it is convenient to use the polar coordinate system forthe spatial discretization. Partition B, when represented by [0,

√b) × [0, 2π], into uniform

rectangles

Ki, j = {(r, θ); r ∈ Ri , θ ∈ � j }, 1 ≤ i ≤ P, 1 ≤ j ≤ Q,

where

ri+ 12

= i�r, θ j+ 12

= j�θ, Ri = [ri− 12, ri+ 1

2], � j = [θ j− 1

2, θ j+ 1

2]

with steps of radius and angle

�r =√

b

P, �θ = 2π

Q.

We define the solution space Vh by

Vh = {v : v|Ki j ∈ Pk(Ki j ), 1 ≤ i ≤ P, 1 ≤ j ≤ Q},where Pk(Ki j ) = span{(r −ri )

p(θ −θ j )q , p +q ≤ k} is the space of polynomials of degree

up to k on Ki j .The DG scheme is as follows: for any function v ∈ Vh , find gh ∈ Vh such that

2∫

Ki, j

M∂t ghv dm = −∫

Ki, j

M(∇m gh − 2κamgh) · ∇mv dm −∫

∂Ki, j

2Mghvκam · �ν ds

(3.4)

+∫

∂Ki, j

M[∇m ghv + (gh − gh)∇mv

]· �ν ds,

where �ν is the outward normal vector on ∂Ki, j .We use a weighted L2 projection to prepare the initial data for the scheme: obtain a

piecewise polynomial gh(m, 0) ∈ Vh , such that in each cell Ki, j ,∫

Ki, j

Mgh(m, 0)v(m) dm =∫

Ki, j

Mg0(m)v(m) dm ∀v ∈ Vh .

Using the polar coordiates in the last term of (3.4), the scheme can be written as

2∫

Ki, j

M∂t ghv dm = −∫

Ki, j

M(∇m gh − 2κamgh) · ∇mv dm − 2∫

∂Ki, j

Mghvκam · �ν ds

+∫

� j

r M[∂r ghv + (gh − gh)∂rv

]∣∣∣r

i+ 12

ri− 1

2

dθ

+∫

Ri

M

r

[∂θ ghv + (gh − gh)∂θv

]∣∣∣∣

θj+ 1

2

θj− 1

2

dr.

123

818 J Sci Comput (2015) 62:803–830

As to the numercial flux, we choose

∂r gh = β0r

�r[gh]+∂r gh +β1r�r [∂2

r gh], ∂θ gh = β0θ

�θ[gh]+∂θ gh +β1θ�θ [∂2

θ gh]. (3.5)

And for gh , we have two options.

(1) The central flux:gh = gh . (3.6)

(2) The upwinding flux:

gh ={

g+h if a > 0,

g−h if a < 0.

(3.7)

At r 12

and rP+ 12, we enforce that g+

h = g−h . This ensures the zero boundary contribution and

incorporates the zero flux boundary condition. Also it is natural to define the second flux in(3.5) periodically.

Using the notation {i j} = {1 ≤ i ≤ P, 1 ≤ j ≤ Q}, the primal form is given by

2∑

i j

∫

Ki, j

M∂t ghv dm + A(gh, v) = 0 ∀v ∈ Vh, (3.8)

where

A(gh, v) =∑

i j

∫

Ki, j

M∇m gh · ∇mv dm

− 2∑

i j

∫

Ki, j

Mκamgh · ∇mv dm + 2∑

i j

∫

∂Ki, j

Mghvκam · �ν ds

+∑

1≤i≤P−11≤ j≤Q

∫

� j

r M(∂r gh[v] + [gh]∂rv

)∣∣∣r

i+ 12

dθ

+∑

i j

∫

Ri

M

r

(∂θ gh[v] + [gh]∂θv

)∣∣∣∣θ

j+ 12

dr.

3.2 Numerical Flux and Coercivity of A

The results from the one dimensional case can be applied to r and θ respectively. We definethe discrete energy norms of v ∈ Vh in r and θ directions as follows:

‖v(·, θ)‖2r =

P∑

i=1

∫

Ri

r M |∂rv|2 dr +P−1∑

i=1

β0r

�rr M[v]2

∣∣∣∣r

i+ 12

,

‖v(r, ·)‖2θ =

Q∑

j=1

∫

�i

M |∂θv|2 dθ +Q−1∑

j=1

β0θ

�θM[v]2

∣∣∣∣θ

j+ 12

.

We introduce two quantities, one at the interface ri+ 12

along the r direction:

�r (β1r , wr ) := supu∈Pk−1([−1,1])

(u(1) − 2β1r∂x u(1))2

∫ 1−1 wr (x)u2(x) dx

,

123

J Sci Comput (2015) 62:803–830 819

the other one at the interface θ j+ 12

along the θ direction:

�θ(β1θ , wθ ) := supu∈Pk−1([−1,1])

(u(1) − 2β1θ ∂x u(1))2

∫ 1−1 wθ(x)u2(x) dx

.

Here the weight functions wr for i = 1, . . . , P − 1 are defined by

wr (x) = min

{(

ri + �r

2x

)

M

(

ri + �r

2x, θ

)

,

(

ri+1 − �r

2x

)

M

(

ri+1 − �r

2x, θ

)}

,

and wθ for j = 1, . . . , Q by

wθ(x) = min

{

M

(

r, θ j + �θ

2x

)

, M

(

r, θ j+1 − �θ

2x

)}

.

Now we are able to establish the coercivity of the bilinear operator A.

Theorem 3.1 (Coercivity)With

β0r > 2ri+ 12

max� j

�r M(ri+ 12, θ) and β0θ > 2 max

Ri�θ M(r, θ j+ 1

2), (3.9)

there exists γ ∈ (0, 1) such that

A(v, v) ≥ γ

⎛

⎜⎝

Q∑

j=1

∫

� j

‖v‖2r dθ +

P∑

i=1

∫

Ri

1

r‖v‖2

θ dr

⎞

⎟⎠

−b

4‖κT κ − κκT ‖

∑

i j

∫

Ki, j

Mv2 dm ∀v ∈ Vh . (3.10)

In order to prove this theorem, we need the following lemma.

Lemma 3.2 If (β0r , β1r ) and (β0θ , β1θ ) satisfy (3.9), then there exists γ ∈ (0, 1) such thatfor any v ∈ Vh,

P∑

i=1

∫

Ri

r M |∂rv|2 dr +P−1∑

i=1

r M(∂rv[v] + [v]∂rv)∣∣r

i+ 12

≥ γ ‖gh‖2r , (3.11)

Q∑

j=1

∫

� j

M |∂θv|2 dθ +Q−1∑

j=1

M (∂θ v[v] + [v]∂θv)

∣∣∣θ

j+ 12

≥ γ ‖v‖2θ . (3.12)

Proof We only discuss (3.11) along the r direction, since the argument for (3.12) is analogous.The definition of �r gives that

(2∂rv + β1r�r [∂2r v])2

∣∣r

i+ 12

≤ 4�r

�r

⎛

⎜⎝

∫

Ri

r M |∂rv|2 dr +∫

Ri+1

r M |∂rv|2 dr

⎞

⎟⎠ . (3.13)

Using the Young inequality, we have

P−1∑

i=1

r M(∂r v[v]+[v]∂r v)∣∣r

i+12

≥P−1∑

i=1

r M

(β0r − α

�r[v]2 − �r

4α(2∂r v + β1r �r [∂2

r v])2)∣∣∣∣r

i+ 12

123

820 J Sci Comput (2015) 62:803–830

≥P−1∑

i=1

r Mβ0r − α

�r[v]2

∣∣∣∣r

i+ 12

−�r ri+ 1

2M(ri+ 1

2, θ)

α

⎛

⎜⎝

∫

Ri

r M |∂r v|2 dr +∫

Ri+1

r M |∂r v|2 dr

⎞

⎟⎠.

Note that there exists γ ∈ (0, 1) such that

β0r =2�r ri+ 1

2M(ri+ 1

2, θ)

(1 − γ )2 .

Then let α =2�r r

i+ 12

M(ri+ 1

2,θ)

1−γand we have

β0r − α = γβ0r and�r ri+ 1

2M(ri+ 1

2, θ)

α= 1 − γ

2.

Therefore

P∑

i=1

∫

Ri

r M |∂rv|2 dr +P−1∑

i=1

r M(∂rv[v] + [v]∂rv)∣∣r

i+ 12

≥ γ ‖v‖2r .

��We now turn to the proof of the coercivity of A.

Proof of Theorem 3.1 Let A(v, v) = I+II+III+IV corresponding to each line in the definitionof A with gh replaced by v. By the change of variables, we have

I =∑

i j

∫

Ki, j

M

(

r |∂rv|2 + 1

r|∂θv|2

)

dr dθ.

Combining this with the estimates (3.11) and (3.12), we have

A(v, v) ≥ γ

⎛

⎜⎝

Q∑

j=1

∫

� j

‖v‖2r dθ +

P∑

i=1

∫

Ri

1

r‖v‖2

θ dr

⎞

⎟⎠+ II.

Next, we only need to analyze II. Using integration by parts, we have

II = − 2∑

i j

∫

Ki, j

Mκamv · ∇mv dm + 2∑

i j

∫

∂Ki, j

M vvκam · �ν ds

= −∑

i j

∫

Ki, j

Mκam · ∇mv2 dm + 2∑

i j

∫

∂Ki, j

M vvκam · �ν ds

=∑

i j

∫

Ki, j

v2∇ · (Mκam) dm +∑

i j

∫

∂Ki, j

M(2v − v)vκam · �ν ds

=V + VI.

123

J Sci Comput (2015) 62:803–830 821

Because ∇m M = (2κsm − bmb−|m|2 )M, we can infer that

∇ · (Mκam) = 2mT κsκamM = 1

4mT (κT κ − κκT )mM. (3.14)

Hence

V ≥ −b

4‖κT κ − κκT ‖

∑

i j

∫

Ki, j

Mv2 dm.

Also notice that κam · �ν = 0 on γ1 and γ3 as shown in Fig. 1. So the boundary term VI in IIbecomes

VI =∑

i j

∫

γ2

M(2v − v−)v−κam · �ν ds +∑

i j

∫

γ4

M(2v − v+)v+κam · �ν ds.

Moreover, the outward normal vector �ν on γ4 of Ki, j and γ2 of Ki, j−1 point in oppositedirections. With this fact and the periodicity in θ , we can shift the line integration on γ4 andget

VI =∑

i j

∫

γ2

M(2v − v−)v−κam · �ν ds −∑

i j

∫

γ2

M(2v − v+)v+κam · �ν ds

= − 4a∑

i j

∫

Ri

Mr(v − v)[v] dr.

For v, if we take (3.1), then VI = 0. If we take (3.7), then we always have

VI = 2|a|∑

i j

∫

Ri

Mr [v]2 dr ≥ 0.

Hence either option ensures the coercivity of A. ��For computational purpose, we further estimate �r and �θ .

Fig. 1 Diagram of the 2Dpartition of B

123

822 J Sci Comput (2015) 62:803–830

Lemma 3.3 If the basis {φμ}k−1μ=1 of Pk−1([−1, 1]) is taken as {xμ−1}k

μ=1, then

�r (β1r , wr ) = ρ(H− 1

2r Or H

− 12

r ) (3.15)

and

�θ(β1θ , wθ ) = ρ(H− 1

2θ Oθ H

− 12

θ ), (3.16)

where ρ denotes the largest eigenvalue of the underlying matrix and the four matrices Or =(Orμν), Hr = (Hrμν), Oθ = (Oθμν) and Hθ = (Hθμν) are defined by

Orμν = (1 − 2β1r (μ − 1))(1 − 2β1r (ν − 1)), 1 ≤ μ, ν ≤ k,

Hrμν =1∫

−1

wr (x)xμ+ν−2 dx, 1 ≤ μ, ν ≤ k,

Oθμν = (1 − 2β1θ (μ − 1))(1 − 2β1θ (ν − 1)), 1 ≤ μ, ν ≤ k,

and

Hθμν =1∫

−1

wθ(x)xμ+ν−2 dx, 1 ≤ μ, ν ≤ k.

Proof We prove the conclusion about �r . Represent u as u(x) = ∑kμ=1 aμxμ−1 = �aT φ(x).

Then

(u(1) − 2β1r∂x u(1))2 = �aT Or �a and u2(x) = �aT φ(x)φT (x)�a.

�aT Hr �a =1∫

−1

wr (x)u(x)2dx > 0 ∀�a �= 0,

implying that Hr is both symmetric and positive definite. So is H± 1

2r . Let y = H

12

r �a, and wehave

�aT Or �a = yT H− 1

2r Or H

− 12

r y ≤ ρ(H− 1

2r Or H

− 12

r )|y|2 = ρ(H− 1

2r Or H

− 12

r )�aT Hr �a.

The conclusion about �θ can be proved in a similar manner. ��3.3 Entropy Stability

Our DG scheme (3.8) then has the following properties.

Theorem 3.4 Consider the semidiscrete DG scheme (3.8) with (3.9). It satisfies the followingproperties:

(1) Conservation of mass:∑

i j

∫

Ki, jfh(m, t) dm = ∫

B f0(m) dm ∀t ≥ 0.

(2) For κ normal, the semidiscrete relative entropy

E(t) =∑

i j

∫

Ki, j

Mg2h dm

is nonincreasing in time. More precisely, there exists γ ∈ (0, 1) such that

d

dtE(t) ≤ −γ

(‖gh‖2r + ‖gh‖2

θ

) ≤ 0. (3.17)

123

J Sci Comput (2015) 62:803–830 823

(3) The scheme is entropy stable in the sense that

E(t) ≤ exp

(bT

4‖κT κ − κκT ‖

)

E(0), for t ∈ [0, T ],

Moreover, E(0) ≤ ∫

B Mg20(m) dm.

Proof (1) Taking the test function v = 1 both in (3.8) and the initial projection, we have themass conservation.

(2) Take v = gh in (3.8) and then use the coercivity of A, we are able to show that E(t)is nonincreasing in time.

For general κ , the estimate (3.10) gives

d

dtE(t) ≤ −γ

(‖gh‖2r + ‖gh‖2

θ

)+ b

4‖κT κ − κκT ‖E(t),

which upon integration in time yields the desired stability estimate in (3). As in the onedimensional case, the weighted L2 projection ensures that E(0) ≤ ∫

B Mg20(m) dm. ��

4 Numerical Tests

In this section, a set of numerical examples will be presented for solving the Fokker–Planckequations (1.1). Examples include the FENE model with the singular force field (1.5), forwhich the main difficulty comes from the boundary singularity. Numerical results demonstratethe optimal order of accuracy of the proposed method, as well as the capacity of the methodto capture solution features for large times.

All the numerical results are obtained using normalized f0 and M , so that the numericalmass and the relative entropy of the equilibrium solution will be one.

4.1 One Dimensional Tests

Denote the numerical solution at t = tn by f nh (m) = M(m)gn

h (m), and the exact solution byf (m, tn). When the exact solution is not available, we replace f (m, tn) by a reference solutionf nre f (m) obtained using refined Nref computational cells. To quantify the computational

accuracy, we measure both the L∞ and L2 errors by

(1) L∞ error is given by

maxmc∈S

| f nh (mc) − f n

re f (mc)|, where S is the set of the cell centers for f nre f .

(2) L2 error is given by⎛

⎝∑

Ire f

‖ f n(·) − f nre f (·)‖2

L2(Ire f )

⎞

⎠

1/2

, where Ire f runs over all the reference cells.

An alternative way to quantify the error is to use the weighted norm for gn .

Example 1 (Accuracy test) We illustrate the accuray of the DG scheme (2.16) with ξ = 1subject to the following initial data.

f0(m) =(

1 − m2

b

) 34 b

.

123

824 J Sci Comput (2015) 62:803–830

Table 1 L∞ and L2 error and order of the accuracy of the Pk approximation for Example 1 on a uniformmesh of N cells, b = 36, �t = 0.001, at the final time t = 0.1

k (β0, β1) N = 5 N = 10 N = 20 N = 40Error Error Order Error Order Error Order

0 (1,0) L∞ 9.102e−02 6.593e−02 0.465 3.161e−02 1.060 1.545e−02 1.033

L2 6.797e−02 3.621e−02 0.908 1.854e−02 0.966 9.315e−03 0.993

1 (3,0) L∞ 3.246e−02 9.783e−03 1.730 2.504e−03 1.966 5.657e−04 2.146

L2 1.524e−02 4.296e−03 1.827 1.102e−03 1.963 2.773e−04 1.990

2 (3, 0.1) L∞ 2.480e−03 5.661e−04 2.131 6.945e−05 3.027 8.624e−06 3.010

L2 1.983e−03 3.962e−04 2.324 4.984e−05 2.991 6.161e−06 3.016

3 (3.5, 0.5) L∞ 6.916e−04 3.678e−05 4.233 2.195e−06 4.066 1.222e−07 4.167

L2 3.944e−04 2.076e−05 4.248 1.056e−06 4.297 6.259e−08 4.077

Table 2 The mass and the relative entropy of the P3 approximation for the uniform mesh N = 20, �t = 0.1and b = 36

t 0 1 10 50 100 800 1,000

Mass 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Entropy 3, 029.250 2, 447.520 561.094 18.948 3.193 1.000 1.000

Fig. 2 b = 36,�t = 0.1.

Table 3 The mass and therelative entropy of the P2

approximation for the uniformmesh P = Q = 10.b = 100, κ11 = κ22 = 0, κ12 =−κ21 = 0.5

t 0 1 10 100

Mass 1.000 1.000 1.000 1.000

Entropy 1.000 1.000 1.000 1.000

We take the numerical solution with N = 640 as the reference solution. Table 1 shows botherror and the order of accuracy of the Pk elements. We can see that the error is uniformlyk + 1-th order in both L2 and L∞ norms for polynomials up to degree k = 3.

123

J Sci Comput (2015) 62:803–830 825

Fig. 3 f0(r, θ) =(

1 − r2

b

) 3b4 e

r2b (0.2 cos(2θ)−0.7 sin(2θ))

, b = 100, κ11 = κ22 = 0.0, κ12 = 0.5, κ21 =−0.5. a t = 0. b t = 1. c t = 2. d t = 100

Example 2 (Entropy satisfying property) Consider the initial data:

f0(m) ={

cos(

3π m√b−η

)+ 1, if |m| ≤ √

b − η,

0, if |m| >√

b − η,

where η is a constant in (0,√

b). The use of η is to ensure that g0 = f0M has a compact support,

and as a result the inital projection has an optimal order of accuracy. For η =√

b5 , Table 2

shows that the numerical solutions preserve the mass and get very close to the discrete steadystate before t = 800, and Fig. 2 illustrates how the numerical solution approaches the steadystate as time evolves. Here we have taken (β0, β1) = (3.5, 0.5) and �t = 0.1.

4.2 Two Dimensional Tests

For two-dimensional problems, we will demonstrate the entropy satisfying property of thefully implicit time discretization of (3.8) with fluxes (3.5) and (3.1). Since (β0, β1) dependon both κ and b, we will give the choice of these pairs for each example to follow. All thesechoices satisfy either the sufficient condition (3.9) or something weaker than (3.9).

Example 3 (Equilibrium preserving) In this example, we consider κ antisymmetric, hencenormal, and take the equilibrium M(m) as the initial data. Table 3 shows that the thirdorder DG method has the capacity of preserving the equilibrium in time. Here (β0r , β1r ) =(β0θ , β1θ ) = (6, 0.5) and �t = 0.1.

123

826 J Sci Comput (2015) 62:803–830

Table 4 The mass and the relative entropy of the P2 approximation for the uniform mesh P = Q = 10 and�t = 0.1. b = 100, κ11 = 0, κ12 = −κ21 = 0.5

t 0 1 2 100 1,000

Mass 1.000 1.000 1.000 1.000 1.000

Entropy 686.634 2.135 1.098 1.000 1.000

Fig. 4 f0(r, θ) =⎧⎨

⎩

cos

(

3π r√b−η

)

+ 1.0, if r ≤ √b − η,

0 else.b = 100, α = 0.3. a t = 0. b t = 1. c t = 10. d

t = 100

Example 4 (Entropy satisfying property) We test the entropy satisfying property of thescheme by two fluid velocities. The first test is on a vortex, with κ being antisymmetricand therefore normal. Figure 3 shows that the numerical solution approaches the equilibriumfeq(m) as time evolves. In this test, (β0r , β1r ) = (β0θ , β1θ ) = (15, 0.35) and �t = 0.1.Indeed, we can observe this entropy satisfying property quantitatively from Table 4. Thesecond test is on a simple extensional flow, with the velocity field

�v = (αx,−αy),

where α denotes the extensional rate. Then the velocity gradient tensor is

κ = ∇�v =(

α 00 −α

)

.

123

J Sci Comput (2015) 62:803–830 827

Table 5 The mass and the relative entropy of the P2 approximation for the uniform mesh P = Q = 10 and�t = 0.1. b = 100, α = 0.3

t 0 1 10 100 1,000

Mass 1.000 1.000 1.000 1.000 1.000

Entropy 1,180.36 2.53658 1.001 1.001 1.001

Table 6 L∞ and L2 error and order of the accuracy of the Pk approximation for Example 4 at time t = 0.4

on a uniform mesh of N × N cells, f0(r, θ) =(

1 − r2

b

) 34 b

, b = 10, �t = 0.0001. Nre f = 80

k (β0, β1) N = 5 N = 10 N = 20

Error Error Order Error Order

0 (1,0) L∞ 3.372e−01 1.622e−01 1.056 6.900e−02 1.233

L2 8.012e−02 4.034e−02 0.990 1.975e−02 1.031

1 (2,0) L∞ 2.379e−01 1.128e−01 1.077 8.284e−02 0.445

L2 1.981e−02 5.432e−03 1.867 1.359e−03 1.999

2 (2, 0.1) L∞ 6.426e−02 6.210e−03 3.371 4.070e−04 3.931

L2 5.209e−03 5.332e−04 3.288 5.892e−05 3.178

Fig. 5 f0(r, θ) =(

1 − r2

b

) 3b4 e

r2b (0.5 cos(2θ)−0.7 sin(2θ))

, b = 100, the final time t = 0.4. a t = 0. b

γ = 0.05. c γ = 0.2. d γ = 0.5

Figure 4 shows that the numerical solution approaches the equilibrium solution feq(m)

as time evolves. While the entropy satisfying property is shown quantitatively in Table 5 forα = 0.3. Here we have taken (β0r , β1r ) = (β0θ , β1θ ) = (15, 0.3) and �t = 0.1.

Example 5 (The simple shear flow) In this example, we test the short time numerical perfor-mance for a simple shear flow, where the gradient of the velocity is

κ = ∇�v =(

0 γ

0 0

)

.

Here we show the numerical convergence by taking b = 10, see the results in Table 6.

Finally we point out that if b becomes larger, M tends to be flatter near the boundary,making it more difficult to evaluate g = f

M . Nevertheless, we present a numerical comparisonin terms of shear rates for b = 100. Figure 5 shows that when the shear rate γ gets larger, the

123

828 J Sci Comput (2015) 62:803–830

two concentrations of f are stretched apart further. Here we have taken k = 1, (β0r , β1r ) =(β0θ , β1θ ) = (6, 0) and �t = 0.01.

5 Concluding Remarks

We have investigated the Fokker–Planck equation which is of bead-spring type FENE dumb-bell model for polymers, with our focus on the development of the entropy satisfying discon-tinuous Galerkin (ESDG) method for the model subject to the zero flux boundary condition.We constructed simple, easy-to-implement ESDG schemes which preserve equilibrium solu-tions. Both semidiscrete and fully discrete methods are proven rigorously to satisfy the desiredproperties: mass conservation and entropy satisfying in the sense that these schemes satisfydiscrete entropy inequalities for the quadratic entropy. We also proved the convergence ofnumerical solutions to the equilibrium solution as time becomes large. Numerical examplesare given to illustrate the accuracy and capability of the methods. The main advantage ofusing the DDG type of numerical fluxes is that there is a room for choosing proper pairs of(β0, β1) so that the desired properties such as the entropy satisfying property and mass con-servation are still preserved for DG methods of an arbitrary order. At the final time, a simplepositive truncation is proposed to approximate the underlying probability density function,and such an approximation is shown as accurate as the obtained numerical solution. Forthis problem, maximum-principle-satisfying third order discontinuous Galerkin schemes forFokker–Planck equations have been recently developed in [45].

Acknowledgments This research was partially supported by the National Science Foundation under GrantDMS09-07963 and DMS13-12636

References

1. Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.: A new family of solvers for some classes of mul-tidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J.Non-Newton. Fluid Mech. 139(3), 153–176 (2006)

2. Ammar, A., Mokdad, B., Chinesta, F., Keunings, R.: A new family of solvers for some classes of multidi-mensional partial differential equations encountered in kinetic theory modelling of complex fluids Part II:Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech. 144(2–3),98–121 (2007)

3. Arnold, A., Carrillo, J.A., Manzini, C.: Refined long-time asymptotics for some polymeric fluid flowmodels. Commun. Math. Sci. 8(3), 763–782 (2010)

4. Arnold, A., Markowich, P., Toscani, G., Unterreiter, A.: On convex Sobolev inequalities and the rate ofconvergence to equilibrium for Fokker–Planck type equations. Commun. Partial Differ. Equ. 26(1–2),43–100 (2001)

5. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer.Anal. 19(4), 742–760 (1982)

6. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methodsfor elliptic problems. SIAM J. Numer. Anal. 39(5):1749–1779, (2001/02)

7. Baker, G.A.: Finite element methods for elliptic equations using nonconforming elements. Math. Comput.31, 45–59 (1977)

8. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solutionof the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)

9. Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems.Comput. Methods Appl. Mech. Eng. 175(3–4), 311–341 (1999)

10. Buet, C., Dellacherie, S.: On the Chang and Cooper scheme applied to a linear Fokker–Planck equation.Commun. Math. Sci. 8(4), 1079–1090 (2010)

123

J Sci Comput (2015) 62:803–830 829

11. Buet, C., Dellacherie, S., Sentis, R.: Numerical solution of an ionic Fokker–Planck equation with electronictemperature. SIAM J. Numer. Anal. 39(4), 1219–1253 (2001). (electronic)

12. Barrett, J.W., Süli, E.: Finite element approximation of finitely extensible nonlinear elastic dumbbellmodels for dilute polymers. ESAIM M2AN 46, 949–978 (2012)

13. Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuousGalerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000). (electronic)

14. Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys.6(1), 1–16 (1970)

15. Chauvière, C., Lozinski, A.: Simulation of complex viscoelastic flows using Fokker–Planck equation: 3DFENE model. J. Non-Newton. Fluid Mech. 122(1–3), 201–214 (2004)

16. Chauvière, C., Lozinski, A.: Simulation of dilute polymer solutions using a Fokker–Planck equation. J.Comput. Fluids 33(5–6), 687–696 (2004)

17. Cheng, Y., Shu, C.-W.: A discontinuous Galerkin finite element method for time dependent partial dif-ferential equations with higher order derivatives. Math. Comput. 77(262), 699–730 (2007)

18. Chupin, L.: The FENE model for viscoelastic thin film flows. Methods Appl. Anal. 16(2), 217–261 (2009)19. Chupin, L.: The FENE viscoelastic model and thin film flows. C. R. Math. Acad. Sci. Paris 347(17–18),

1041–1046 (2009)20. Chupin, L.: Fokker–Planck equation in bounded domain. Ann. Inst. Fourier (Grenoble) 60(1), 217–255

(2010)21. Cockburn, B., Dawson, C.: Some extensions of the local discontinuous Galerkin method for convection-

diffusion equations in multi-dimensions. In: Whiteman, J.R. (ed.) Proceedings of the Conference on theMathematics of Finite Elements and Applications, MAFELAP X, pp. 225–238. Elsevier, New York (2000)

22. Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finiteelement method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581(1990)

23. Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuousGalerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)

24. Cockburn, B., Lin, S.Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finiteelement method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84(1), 90–113(1989)

25. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws. II. General framework. Math. Comput. 52(186), 411–435 (1989)

26. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

27. Du, Q., Liu, C., Yu, P.: FENE dumbbell model and its several linear and nonlinear closure approximations.Multiscale Model. Simul. 4(3), 709–731 (2005). (electronic)

28. Fan, X.-J.: Viscosity, first normal-stress coefficient, and molecular stretching in dilute polymer solutions.J. Non-Newton. Fluid Mech. 17(2), 125–144 (1985)

29. Gassner, G., Lörcher, F., Munz, C.-D.: A contribution to the construction of diffusion fluxes for finitevolume and discontinuous galerkin schemes. J. Comput. Phys. 224(2), 1049–1063 (2007)

30. Herrchen, M., Öttinger, H.C.: A detailed comparison of various FENE dumbbell models. J. Non-Newton.Fluid Mech. 68(1), 17–42 (1997)

31. Hyon, Y., Du, Q., Liu, C.: An enhanced macroscopic closure approximation to the micro-macro FENEmodel for polymeric materials. Multiscale Model. Simul. 7(2), 978–1002 (2008)

32. Hesthaven, J., Warburton, T.: Nodal Discontinuous Galerkin Methods, Algorithms, Analysis and Appli-cations. Springer, Berlin (2008)

33. Jourdain, B., Le Bris, C., Lelièvre, T., Otto, F.: Long time asymptotics of a multiscale model for polymericfluid flows. Arch. Ration. Mech. Anal. 181(1), 97–148 (2006)

34. Jourdain, B., Lelièvre, T.: Mathematical analysis of a stochastic differential equation arising in the micro-macro modelling of polymeric fluids. In: Probabilistic Methods in Fluids, pp. 205–223. World Sci. Publ.,River Edge, NJ (2003)

35. Jourdain, B., Lelièvre, T., Le Bris, C.: Existence of solution for a micro-macro model of polymeric fluid:the FENE model. J. Funct. Anal. 209(1), 162–193 (2004)

36. Knezevic, D.J., Süli, E.: Spectral Galerkin approximation of Fokker–Planck equations with unboundeddrift. M2AN Math. Model. Numer. Anal. 43(3), 445–485 (2009)

37. Li, B.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer, London (2006)38. Larsen, E.W., Levermore, C.D., Pomraning, G.C., Sanderson, J.G.: Discretization methods for one-

dimensional Fokker–Planck operators. J. Comput. Phys. 61(3), 359–390 (1985)39. Le Bris, C., Lelièvre.: Micro-macro models for viscoelastic fluids: modeling, mathematics and numerics.

arXiv:1102.0325v1[math-ph] (2011)

123

830 J Sci Comput (2015) 62:803–830

40. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J.Numer. Anal. 47(1), 675–698 (2009)

41. Liu, H., Shin, J.: The Cauchy–Dirichlet problem for the FENE dumbbell model of polymeric flows. SIAMJ. Math. Anal. 44(5), 3617–3648 (2012)

42. Liu, H., Shin, J.: Global well-posedness for the microscopic FENE model with a sharp boundary condition.J. Differ. Equ. 252(1), 641–662 (2012)

43. Liu, H., Yan, J.: The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections.Commun. Comput. Phys. 8(3), 541–564 (2010)

44. Liu, H., Yu, H.: An entropy satisfying conservative method for the Fokker–Planck equation of FENEdumbbell model for polymers. SIAM J. Numer. Anal. 50(3), 1207–1239 (2012)

45. Liu, H., Yu, H.: Maximum-principle-satisfying third order discontinuous Galerkin schemes forFokker–Planck equations. SIAM J. Sci. Comput. (2013). http://www.ams.org/amsmtgs/2216_abstracts/1097-65-286.pdf

46. Lozinski, A., Chauvière, C.: A fast solver for Fokker–Planck equation applied to viscoelastic flowscalculations: 2D FENE model. J. Comput. Phys. 189(2), 607–625 (2003)

47. Masmoudi, N.: Well-posedness for the FENE dumbbell model of polymeric flows. Commun. Pure Appl.Math. 61(12), 1685–1714 (2008)

48. Masmoudi, N.: Global existence of weak solutions to the FENE dumbbell model of polymeric flows.Invent. Math. 191(2), 427–500 (2013)

49. Markowich, P.A., Villani, C.: On the trend to equilibrium for the Fokker–Planck equation: an interplaybetween physics and functional analysis. Phys. Funct. Anal. Mat. Contemp. (SBM) 19, 1–29 (1999)

50. Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems.J. Comput. Phys. 146(2), 491–519 (1998)

51. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Technical ReportTech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, (1973)

52. Riviére, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, Theory andImplementation, SIAM (2008)

53. Samaey, G., Lelièvre, T., Legat, V.: A numerical closure approach for kinetic models of polymeric fluids:exploring closure relations for FENE dumbbells. Comput. Fluids 43, 119–133 (2011)

54. Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In Bertoluzza, S., Falletta,S., Russo, G., Shu, C.-W. (eds.) Numerical Solutions of Partial Differential Equations, pp. 149–201,Birkhauser Basel (2009)

55. Shen, J., Yu, H.J.: On the approximation of the Fokker–Planck equation of FENE dumbbell model, I: anew weighted formulation and an optimal Spectral-Galerkin algorithm in 2-D. SIAM J. Numer. Anal.50(3), 1136–1161 (2012)

56. van Leer, B., Nomura, S.: Discontinuous Galerkin for diffusion. In: Proceedings of 17th AIAA Compu-tational Fluid Dynamics Conference (6 June 2005), AIAA-2005-5108 (2005)

57. Wang, H., Li, K., Zhang, P.: Crucial properties of the moment closure model FENE-QE. J. Non-Newton.Fluid Mech. 150(2–3), 80–92 (2008)

58. Warner, H.R.: Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind.Eng. Chem. Fundam. 11(3), 379–387 (1972)

59. Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer.Anal. 15, 152–161 (1978)

60. Zhang, H., Zhang, P.: Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration.Mech. Anal. 181(2), 373–400 (2006)

123