Numerical methods for computing an averaged matrix field. Application to the asymptotic analysis of a parabolic problem with stiff transport terms

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Numerical methods for computing an averaged matrix field. Application to the asymptotic analysis of a

parabolic problem with stiff transport terms

Thomas Blanc

To cite this version:

Thomas Blanc. Numerical methods for computing an averaged matrix field. Application to the asymptotic analysis of a parabolic problem with stiff transport terms. 2017. �hal-01599005�

Numerical methods for computing an averaged matrix field.

Application to the asymptotic analysis of a parabolic problem with stiff transport terms.

Thomas Blanc ^∗ (July 20, 2017)

Abstract

Parabolic problems with stiff terms are challenging to solve numerically. When the stiff terms become dominant, multiple scale effects occur and the classical numerical methods do not catch the microscopic effects. The aim of this paper is to provide a numerical method to study the behavior of parabolic problems with stiff transport terms based on recent results on the asymptotic analysis for such problems. Precisely, the behavior of the solutions can be described in terms of a composition product between a certain profile and the fast flow associated to the dominant transport operator, where the asymptotic profile solves an effective diffusion equation. A numerical method for determining the effective diffusion matrix is given, the computation of the limit profile is carried out and the error with respect to the solution of the stiff problem is studied.

Keywords: Averaging methods, Semi-Lagrangian schemes, Multiple scales.

AMS classification: 65M25, 65M06.

1 Introduction

In many applications, partial differential equations with multiple scales can occur : transport in strongly magnetized plasmas with or without collisions, heat transfer inside the plasma fusion, heat and mass transport in the chemical framework. Each of these problems makes appear multiple scales in time or space. From the numerical point of view, the study of these problems is highly constraint by the size of the small scale. Indeed, the numerical resolution must be thin enough for catching the effects caused by the small scales. But, in this case, the classical methods have a prohibitive numerical cost and are not adapted for solving this type of problems. In this work we provide a way to study numerically the behavior of a diffusion equation with a stiff convection term. We focus on the following parabolic model

#

Btu^ε´divypDpyq∇_yu^εq `1

ε bpyq ¨∇_yu^ε“0, pt, yq PR`ˆR^m u^εp0, yq “uⁱⁿpyq, yPR^m,

(1.1) where b : R^m Ñ R^m and D : R^m Ñ M_mpRq are given fields of vectors and symmetric positive definite matrices, and εą0 is a small parameter close to zero. The fast transport, which is related to the operator b¨∇_y, introduce a fast time scale. Actually, this problem

∗Aix Marseille Universit´e, CNRS, Centrale Marseille, Institut de Math´ematiques de Marseille, UMR 7373, Chˆateau Gombert 39 rue F. Joliot Curie, 13453 Marseille FRANCE. E-mail : [email protected]

can be interpreted as a two-scale problem in time, after a convenient Lagrangian change of variable along the fast motion. Multiple-scale problems have been extensively studied by many authors and there exists a lot of approaches for their numerical study. The strongly anisotropic diffusion problems have been analyzed by using asymptotic preserving schemes [16], the kinetic equations [12], the multiple-scale parabolic problems [13], the Schr¨odinger and Klein-Gourdon equations [10] have been adressed by appealing to uniform accurate schemes.

Multiple-scale methods for advection-diffusion equations are proposed in [1]. The strategy we choose in this paper will be not to solve numerically the problem (1.1), but rather an homogenized limit problem not constraint by the small parameterε, and for which standard numerical solvers can be used. Indeed, the theoretical study of multiple scales problems by homogenization techniques has been done in many frameworks such as transport with disparate advection fields [4, 9, 15], transport of charged particles under high magnetic fields [?, 6, 7, 8, 18, 19], elliptic and parabolic models [2, 22] or asymptotic analysis of strongly anisotropic diffusion problems [8]. In the recent paper [3], the asymptotic analysis of the problem (1.1) has been performed through an ergodic theory result. It has been shown that the behavior of the family pu^εqεą0, when ε goes to zero, can be described, in the L² sense, with a convergence rate, in terms of the composition product between a profile solution of the homogenized problem and the fast oscillating flow associated to b{ε. Moreover, it is shown that the homogenized problem is still a parabolic problem, whose effective diffusion matrix field is given by an ergodic average of the initial diffusion matrix fieldD, along a group of linear operators. Actually the infinitesimal generator associated to this group is a transport operator which acts on matrix fields. The main goal of this article is to provide a semi-Lagrangian scheme for solving the group, and thus compute the effective diffusion matrix field. The interest of this numerical computation is not restricted to the problem (1.1). Indeed, the average of a matrix field is found in various situations and makes it possible to describe, for example, the effective system associated with a strongly anisotropic diffusion equation, cf.

[8]. We illustrate this approach by observing the error between a reference solution of (1.1) and the solution of the effective diffusion problem for some particular examples.

This paper is organized as follow. In Section 2 we introduce the notations which will be used through this study and we recall the main asymptotic results established in [3]. A scheme based on a semi-Lagrangian method is provided in Section 3 for the computation of the effective diffusion field. Some numerical tests are done as well. In Section 4, we study the error between the solution of the effective problem, with respect to the solution of the stiff problem (1.1). We use a numerical scheme based on splitting methods for providing a reference solution for (1.1). The numerical results confirm the expected theoretical convergence rates.

2 Asymptotic analysis, theoretical results

In this section we introduce some notations and results which will be useful along the paper.

The main points are the definition of the effective diffusion matrix field (2.4), (2.1) and the asymptotic result (2.6). We only indicate the main lines of the arguments leading to these results. For the proof details we refer to [3]. Consider Y :RˆR^m Ñ R^m the characteristic flow of the vector field b

dY

ds “bpYps;yqq, ps, yq PRˆR^m, Yp0;yq “y, y PR^m.

This flow is well defined under the standard smoothness and boundedness assumptions

#bPW_loc^1,8pR^mq,

DCą0 such that|bpyq| ďCp1` |y|q, y PR^m.

Under the above hypotheses the flowY is global and smooth,Y PW_loc^1,8pRˆR^mq. We assume that the vector fieldb is divergence free

div_yb“0,yPR^m

which guarantees that the transformation y PR^m ÑYps;yq PR^m is measure preserving for anysPR. The asymptotic analysis of (1.1) comes immediately when the operatorsb¨∇_y and div_ypD∇_yq are commuting, i.e. rb¨∇_y,div_ypD∇_yqs “0. The idea is to perform the change of coordinates z“Yp´t{ε;yq, and therefore to replace the familypu^εqεą0 by the new family pv^εqεą0 given by

u^εpt, yq “v^εpt, zq “v^εpt, Yp´t{ε;yqq, pt, yq PR`ˆR^m.

The point is that, under the above commutation property, the new unknownspv^εqεą0 satisfy

"

Btv^ε´div_zpD∇_zv^εq “0, pt, zq PR`ˆR^m v^εp0, zq “uⁱⁿpzq, zPR^m.

Thus, v^ε does not depend on ε and therefore u^ε is the composition between the profile v “ v^ε and the flow associated to the vector field b{ε. The matrix fields D which ensure the commutation property are characterized in the following proposition, see [8, 3] for more details.

Proposition 2.1 Consider a divergence free vector field cPW_loc^1,8pR^mq with at most linear growth at infinity and APL¹_locpR^m,M_mpRqq a matrix field.

1. The commutator between the advection operatorc¨∇_y and the diffusion operatordiv_ypA∇_yq is still a diffusion operator, and we have

“c¨∇_y,div_ypA∇_yq‰

“div_yprc, As∇_yq in D¹pR^mq

where the associated diffusion field is defined by the bracket between the vector field c and the matrix field A

rc, As:“ pc¨∇_yqA´ BycA´A^tByc in D¹pR^mq.

2. The following assertions are equivalent (a) We have rc, As “0 in D¹pR^mq

(b) For any sPR, we have GpsqA“A, where the family of linear operators pGpsqq_s, acting on matrix fields, is defined by

pGpsqAqpyq:“ BY^´1ps;yqApYps;yqq^tBY^´1ps;yq, ps, yq PRˆR^m. (2.1) Motivated by the case in which the operators commute, we perform the asymptotic analysis not for the family pu^εqεą0 but rather for the new family of functionspv^εqεą0 given by

v^εpt, zq “u^εpt, Ypt{ε;zqq, pt, zq PR`ˆR^m, εą0.

We expect that the familypv^εqεą0converges whenεgoes to zero. Actually, when the operators b¨∇_y and div_ypD∇_yq are not commuting, the asymptotic behavior of the family pu^εqεą0, when εgoes to zero, can be described asymptotically in the same way as before, that is as a composition product between a profile v and the flow associated to b{ε. This profile v is the solution of a diffusion equation with a new diffusion matrix field xDy, which appears as

the orthogonal projection of Dover the linear space of matrix fields which are left invariant by the family pGpsqq_s in (2.1), with respect to some scalar product, to be precised. The definition ofxDyis given by (2.4). Performing the change of variablez“Y p´t{ε;yqin (1.1), for anyεą0,tPR`,yPR^m, and appealing to the chain rule (see [3] for more details) lead to a two time scales diffusion problem

"

Btv^ε´div_zppGpt{εqDq∇_zv^εq “0, pt, zq PR`ˆR^m

v^εp0, zq “u^εp0, zq “uⁱⁿpzq, zPR^m, εą0 (2.2) where pGpsqq_sPR is defined by (2.1). A two-scale approach, based on Hilbert’s method, for- mally leads to the following effective problem

"

Btv´div_zpxDy∇_zvq “0, pt, zq PR`ˆR^m

vp0, zq “uⁱⁿpzq, zPR^m, (2.3)

where the effective diffusion fieldxDyis given by the long time average xDy “ lim

SÑ`8

1 S

ż_S

0

GpsqDds. (2.4)

Remark 2.1 The computation of such average matrix field does not depend on the initial conditionuⁱⁿ. Thus, a pre-computation of the average matrix field is possible by knowing only D and b.

The existence of the matrix field xDy is provided by the von Neumann ergodic theorem, see [23, 3]. Indeed, the family of linear operators pGpsqq_sPR defined by (2.1) is a C⁰-group of unitary operators in a suitable Hilbert space H_Q, see (3.12). The existence of such Hilbert space is not always ensured, thus the average matrix field (2.4) is not always defined in this sense, see Section 3.4 or [3] for more details. The average matrix field (2.4) can be interpreted as a projection on the kernel of the infinitesimal generator L associated to pGpsqq_sPR. We have a description for the restriction of L to the compactly supported smooth matrix fields as a transport operator

LpAq “ rb, As “ pb¨∇_yqA´ BybA´A^tByb, APC_c¹pR^mq. (2.5) This expression of L will be useful for the computation of the group pGpsqq_s, see Section 3.

Now, we can describe the behavior of the family pu^εqεą0 whenε goes to zero. It is shown in [3] that, for any initial condition uⁱⁿ PL²pR^mq, the family pv^εqεą0 solutions of the equation (2.2) converges strongly inL⁸_locpR`;L<sup>2</sup>pR<sup>m</sup>qq, to the unique solution vPL<sup>8</sup>pR`;L²pR^mqqof (2.3). Actually, if we consider T ą0, there is a constant C_T ą0 such that, for anyεą0, we have

sup

tPr0,Ts

}v^εpt,¨q ´vpt,¨q}_L²_pR^m_qďC_Tε and

sup

tPr0,Ts

}u^εpt,¨q ´vpt, Yp´t{ε;¨qq}_L²_pR^m_qďC_Tε. (2.6)

3 Computation of the average matrix field

In this section, we provide a numerical method for the computation of the average matrix field (2.4). IfD : R^m ÑM_mpRqis a matrix field, the computation of the average matrix field xDydefined by (2.4) will be provided in two steps. First, we compute numerically the matrix field GpsqD “ pGpsqDqpyq by using the infinitesimal generator L given by (2.5). Secondly,

the computation of the long time average will be done by a quadrature method. The method is presented in Section 3.1 and the numerical scheme associated is performed in Section 3.2.

In Section 3.3, the accuracy of the method is tested for several vector fields b. The cases where the flow associated to the vector fieldbis known provide, thanks to the formula (2.1), reference curves for the one parameter grouppGpsqq_sPR. Finally, it seemed interesting to give an example of vector fieldb for which the associated average matrix field is not well defined, we propose such example in Section 3.4.

3.1 Computation of the matrix field GpsqD

Consider S ą0 and DPC_c¹pR^m,M_mpRqq a compactly supported smooth matrix field. The matrix field pGpsqDqpyq where sP r0, Ss, y PR^m is related to the infinitesimal generatorL of the grouppGpsqq_sPR through the following evolution problem

$

’&

’% d

dspGpsqDqpyq “LpGpsqDq pyq, ps, yq P r0, Ss ˆR^m Gp0qDpyq “Dpyq, yPR^m.

Moreover, the operator Lis a transport operator and its explicit expression is given in terms ofband its partial derivatives with respect toy, see (2.5). Thus, we need to solve the following partial differential equation on the matrix fieldAps, yq:“ pGpsqDqpyq,ps, yq P r0, Ss ˆR^m

$

’&

’% d

dsAps, yq “ pb¨∇_yqAps, yq ´ B_ybpyqAps, yq ´Aps, yq^tBybpyq, ps, yq P r0, Ss ˆR^m

Ap0, yq “Dpyq, yPR^m.

(3.1) For the resolution of this equation, we appeal to a semi-Lagrangian scheme, see [17]. Such schemes have several advantages : they are unconditionally stable, with arbitrary order of accuracy and are known to provide less numerical diffusion that the Eulerian schemes, as upwind schemes. To avoid a too prohibitive numerical cost, we are not allowed to choose a small resolution. It is the reason why, we choose a high order of accuracy (four in practice) for the computation of the diffusion matrix field GpsqDforsP r0, Ss. As the long time compu- tation of GpsqDis required, we need that the scheme be not too diffusive. All these remarks have guided us to provide a numerical scheme based on the semi-Lagrangian framework for the resolution of (3.1). We rewrite the equation (3.1) by using the Lagrangian change of coordinates y“Yp´s;zq for ps, zq P r0, Ss ˆR^m. By setting Bps, zq:“Aps, Yp´s;zqq, we have

$

’&

’% d

dsBps, zq “ ´BybpYp´s;zqqBps, zq ´Bps, zq^tBybpYp´s;zqq, ps, zq P r0, Ss ˆR^m

Bp0, zq “Dpzq, yPR^m.

(3.2) We solve the ordinary differential equation (3.2) by a Runge-Kutta scheme of order four.

The unknown change Aps₀, yq “Bps₀, Yps₀;yqq, fors₀ P r0, Ss, y P R^m, is performed by a semi-Lagrangian scheme. The flow Yps₀;yq is computed by a Runge-Kutta solver, and the reconstruction ofBps0, Yps0;yqq, from the values ofBps0,¨qon the grid points, is achieved by Lagrangian interpolation. This last step can be interpreted as the resolution of the following

transport equation

$

’&

’% d

dsCps, yq “ pb¨∇_yqCps, yq, ps, yq P r0, Ss ˆR^m Cp0, yq “Bps₀, yq, yPR^m.

(3.3)

Indeed, if we use the method of characteristics method on (3.3), we obtain Cps₀, yq “ Bps0, Yps0;yqq “Aps0, yq. The construction of the scheme is detailed in Section 3.2.

3.2 Numerical scheme for the computation of xDy

For simplicity, the scheme is presented in the two dimensional setting m“2, but it extends easily to any dimension m ě 3. Several examples are given for m “4 in Section 3.3. The spatial domain will be a squareC “ r´R, Rs ˆ r´R, Rs, forRą0. ConsiderS ą0,N_sPN^‹ andI_s“ t0, . . . , N_su. We introduce a regular discretizationps^kqkPI_sof the intervalr0, Sswith a step ∆s“S{Ns, thuss^k“k∆sfor anykPI_s. Moreover, we defineds<sup>k`1{2</sup> “ pk`1{2q∆s for any kP t0, . . . , N_s´1u. ConsiderN PN^‹ and I “ t0, . . . , Nu. Assume that py_ijqpi,jqPI²

is a regular cartesian discretization of the square C, with i, j P I. The step of the spatial discretization is ∆y “2R{N, and yij “ p´R`i∆y, ´R`j∆yq for any pi, jq PI².

Resolution of the system (3.2)

For the approximation of Bps^k, yijq, for k P I_s and pi, jq P I², we introduce the matrices B^k_ij P M₂pRq. We solve numerically the system (3.2) at any point y_ij, i, j P I, with a Runge-Kutta 4 solver. We define the application F_yij : RˆM₂pRq Ñ M₂pRq, ps, Mq ÞÑ

´BybpYp´s;yijqqM ´M^tBybpYp´s;yijqq. The scheme writes B_ij⁰ “

ˆD11pyijq D12pyijq D21pyijq D22pyijq

˙

B_ij<sup>k`1</sup> “B<sub>ij</sub><sup>k</sup> `∆s

6 pk1,ij`2k2,ij`2k3,ij`k4,ijq (3.4) where

k_1,ij “F_yijps^k, B_ij^kq

k_2,ij “F_yij ˆ

s<sup>k`1{2</sup>, B<sub>ij</sub><sup>k</sup> `∆s 2 k_1,ij

˙

k3,ij “F_yij ˆ

s<sup>k`1{2</sup>, B<sub>ij</sub><sup>k</sup> `∆s 2 k2,ij

˙

k_4,ij “F_yij´

s<sup>k`1</sup>, B<sub>ij</sub><sup>k</sup> `∆sk_3,ij

¯ .

At each step of the scheme, the computation of the flow Yps;¨qassociated to the vector field bp¨q needs to be evaluated at times s“ ´s^k,s“ ´s^{k`1{2</sup> and s“ ´s<sup>k`1}.

Semi-Lagrangian step

We introduce the matrices A^k_ij P M₂pRq for the approximation of the matrices Aps^k, yijq for k P I_s and pi, jq P I². We know that Aps^k, y_ijq “ Bps^k, Yps^k;y_ijqq, we first compute the flow Yps;¨q at time s “ s^k, at any point yij, pi, jq P I² by a Runge-Kutta 4 solver.

The approximations B_ij^k of Bps^k, yijq are used for the reconstruction of Bps^k, Yps^k;yijqq by a Lagrangian interpolation. More exactly, cubic Lagrangian interpolation is performed by

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

‚

›

Stencil for the interpolation

‚ Evaluation points

› PointYps^k, y_ijq

Figure 1: Stencil S

using the values of 16 points surrounding Yps^k;y_ijq. We nameSthe associated stencil, see Figure 1 :

More precisely, the interpolation of the function Bps^k,¨qat the pointYps^k;yijq will be done by a polynomial function Ppy1, y2q “ ÿ

0ďα, βď3

aαβy₁^αy₂^β. The computation of the coefficients a_αβ, for 0 ď α, β ď 3, is based on the 16 values of B^k_ij, such that y_ij P S, by solving the linear system associated to the equations

ÿ

0ďα, βď3

a_αβpyijq^α₁pyijq^β₂ “B^k_ij, yij PS.

This Lagrangian interpolation provides a fourth order space approximation for regular data.

Computation of xDy

The computation of xDy is performed by a numerical integration of s ÞÑ GpsqD on r0, Ss.

Actually, a four points Newton-Cotes quadrature method is used, which is five order accurate in time, for more details on these methods see [14]. Consider ˜Dij, fori, j PI, an element of M₂pRqwhich approximatesxDy pyijq. The quadrature method, with the valuesA^k_ij computed in Section 3.2, writes

D˜ij “ 1 S

Ns{4´1

ÿ

k“0

4∆s

4

ÿ

l“0

ωlA^4k`l_ij

where the coefficients of quadrature are given by ω₀ “ω₄ “ 7

90,ω₁ “ω₃ “ 16

45 and ω₂ “ 2 15. 3.3 Numerical computations and examples

In this section, we test the previous numerical method, for computing the average matrix field. LetxDy pyq be the average matrix field associated toDpyq and ˜Dbe the numerical ap- proximation given by the numerical scheme. The error analysis between these two quantities is localized to a domainS ĂCwhich is left invariant by the flow associated to the vector field b, i.eYps;Sq “S for any sPR. For any matrixAPM₂pRq,|A| “?

A:A, we introduce the relative error based on the discreteL² norm

ErrorL²“ g f f e

ř

pi, jqPS| xDy py_ijq ´D˜_ij|² ř

pi, jqPS| xDy pyijq|² . (3.5)

Ellipsoidal flow

We consider the Hamiltonian vector field bpy₁, y₂q “ pBy2Hpyq,´By1Hpyqq, for y “ py₁, y₂q P R², with

Hpyq “ 1 2y₁²`1

2y²₂`1

2y1y2, y“ py1, y2q PR².

The function H is a coercive prime integral associated to b. We denote by Y the flow associated to the vector field b. The characteristic curves are ellipses of the plane, the flow Yps;yq “Epsqy is 4π{?

3-periodic, and we have Epsq “

¨

˝ cos

´? 3 2 s

¯

`^?1₃sin

´? 3 2 s

¯ ?2

3sin

´? 3 2 s

¯

´^?2₃sin

´? 3 2 s

¯

cos

´? 3 2 s

¯

´^?1₃sin

´? 3 2 s

¯

˛

‚.

Thanks to (2.1), we have xDy “

?3 4π

ż_4π{

?3

0

GpsqDds“

?3 4π

ż_4π{

?3

0

Ep´sqDpYps;yqq^tEp´sqds“

ˆ xDy₁₁ xDy₁₂ xDy₂₁ xDy₂₂

˙

where

xDy₁₁“ 1 2

A

D₁₁r1`cosp? 3¨qs

E

´ 1

?3 A

pD₁₁`D<sub>21</sub>`D₁₂qsinp? 3¨q

E

`1 6

A

pD11`2pD21`D12q `4D22qr1´cosp? 3¨qs

E

xDy₂₁“ 1 2

A

D21r1`cosp? 3¨qs

E

` 1

?3 A

pD11´D22qsinp? 3¨q

E

´1 6

A

p2D₁₁`D<sub>21</sub>`4D₁₂`2D₂₂qr1´cosp? 3¨qs

E

xDy₁₂“ 1 2

A

D₁₂r1`cosp? 3¨qs

E

` 1

?3 A

pD₁₁´D₂₂qsinp? 3¨q

E

´1 6

A

p2D₁₁`4D<sub>21</sub>`D₁₂`2D₂₂qr1´cosp? 3¨qs

E

xDy₂₂“ 1 2

A

D₂₂r1`cosp? 3¨qs

E

` 1

?3 A

pD₂₁`D<sub>12</sub>`D₂₂qsinp? 3¨q

E

`1 6

A

p4D11`2pD21`D12q `D22qr1´cosp? 3¨qs

E

and for any function h, we denotexD_ijhp¨qy “

?3 4π

ş_4π{

?3

0 D_ijpYps;yqqhpsq ds, pi, jq P t1,2u². Finally, ifD₁₁, D₂₁, D₁₂, D₂₂are constant along the flowYps;¨q(i.eD₁₁, D₂₁, D₁₂, D₂₂only depend on the quantityy₁²`y<sub>2</sub><sup>2</sup>`y1y2), the explicit expression of xDyreduces to

xDy “ ˆ ₂

3pD₁₁`D<sub>22</sub>q `¹₃pD₂₁`D<sub>12</sub>q ´<sup>1</sup><sub>3</sub>pD<sub>11</sub>`D₂₂q `¹₃pD₁₂´2D₂₁q

´¹₃pD₁₁`D<sub>22</sub>q ` ¹₃pD₂₁´2D₁₂q ²₃pD₁₁`D<sub>22</sub>q `¹₃pD₂₁`D₁₂q

˙

. (3.6) We consider the two following matrix fields

D₁ “ ˆ3 1

2 1

˙

and D₂pyq “

ˆ3`cospκpy<sub>1</sub><sup>2</sup>´y<sub>2</sub><sup>2</sup>qq cospy<sup>2</sup><sub>1</sub>´2y2q sinp2y<sub>1</sub>´y<sub>2</sub>q 3`sinpκpy²₁´y₂qq

˙

withκ“25.

10^´2 10^´1 10^´11

10^´10 10^´9 10^´8 10^´7 10^´6

∆y L2 error

xD₁y

L² error

Slop e3.99

10^´2 10^´1

10^´7 10^´6 10^´5 10^´4 10^´3 10^´2

∆y L2 error

xD₂y

L² error

Slop e3.97

Figure 2: Left : L² error for the approximation of xD₁y. Right : L² error for the approxima- tion of xD₂y.

The explicit expression of xD2ycan be determined by the above formula and the average of D₁ is given by

xD₁y “

ˆ11{3 ´7{3

´4{3 11{3

˙ .

We check that the global space-time error committed for the computation ofxDyisOp∆s⁴q ` Op∆y<sup>4</sup>q. We choose C “ r´1; 1s<sup>2</sup> and ∆s “ ∆y. The Figure 2 shows the relative errors (3.5) committed when using the method presented in Section 3.2. The graphic on the left emphasizes aOp∆y<sup>4</sup>qerror. In this example only the time error appears due to the resolution of the ordinary differential equation (3.2) by the Runge-Kutta 4 scheme (3.4). Indeed, the matrix field D1 is constant and the Jacobian matrix associated to b is also constant. The solution of (3.2) does not depend on the variable y. In this case, the interpolation step presented in Section 3.2 is exact. The graphic on the right presents the relative error in non constant, but regular, case. The interpolation error is orderOp∆y<sup>4</sup>qand the global time-space error is Op∆s<sup>4</sup>q `Op∆y⁴q.

Non explicit central flow

We consider the Hamiltonian vector field bpy₁, y₂q “ pBy2Hpyq,´By1Hpyqq, defined for y “ py1, y2q P R², with H a homogeneous function of degree two, i.e Hpλy1, λy2q “ λ²Hpy1, y2q for any λPR and py1, y2q PR²

Hpy1, y2q “1 2y₁²`1

2y₂²`y1y2py₁²´y²₂q

y₁²`y<sup>2</sup><sub>2</sub> , py1, y2q PR<sup>2</sup>{tp0,0qu, and Hp0,0q “0. (3.7) The functionHisC<sup>2</sup>onR<sup>2</sup>{tp0,0quand is a coercive prime integral ofb. We denote byY the flow associated to the vector fieldb. The characteristic curves are closed and the flow Yps;¨q is periodic. A representation of these curves are given by the Figure 3. In polar coordinates, the function H defined byHprcospθq, rsinpθqq “Hpr, θq, forpr, θq PR`ˆ r0,2πr, writes

Hpr, θq “r² ˆ1

2`gpθq

˙

with gpθq “ 1

4sinp4θq.

Figure 3: Level set of the function (3.7)

The periodTplqassociated with each closed integral curveH “l,lě0 because min_R²H“0, is constant Tplq “T. Indeed, we can computeT, see [20], by the following formula

T “ ż_2π

0

dθ 1`2gpθq “

ż_2π

0

dθ

1`¹₂sinp4θq “ 4π

?3.

The average matrix field associated to a matrix fieldD is given, thanks to (2.1), by xDy “

?3 4π

ż4π{? 3 0

GpsqDds“

ˆ xDy₁₁ xDy₁₂ xDy₂₁ xDy₂₂

˙ .

In this case, we do not have an explicit expression for the flow Y, and thus for the matrix field GpsqD. If the matrix field D is constant, we can obtain some informations on the average matrix field. The Hamiltonian (3.7) is homogeneous of degree two, we deduce that bpλyq “λbpyq, for anyλPRand yPR². Thus, we have

Yps;λyq “λYps;yq, for anyλPRandps, yq PRˆR². (3.8) By differentiation of the equality (3.8) with respect to y, we obtain

BYps, λyq “ BYps, yq, for any λ‰0, yPR².

IfDpλyq “Dpyq, for anyλ‰0 andyPR², then, the formula (2.1) yields : xDy pλyq “ xDy pyq for any λ‰0 and yPR². Thus, we expect that the average with respect tob of a constant matrix field is constant along the straight lines passing through the origin of the plane, with a discontinuity at y “ p0,0q. Indeed, the Hamiltonian H is not C² at the origin, thus the average matrix field xDy is not continuous at the point y “ p0,0q. We test the method of Section 3.2 for the matrix field D “ I₂. We choose C “ r´1; 1s ˆ r´1; 1s as a numerical domain. The Figure 4 represents the coefficients of the matrix field xI₂y localized to an invariant domain with respect to the flow ofb. Each of these coefficients is constant along the straight lines passing to the origin with a discontinuity aty“ p0,0q. In Figure 5, we observe the order of accuracy of the method with ∆s“∆y. The reference solution is obtained by the average computation method and a spatial resolution N “512. The default of regularity at the origin causes a degradation of the order of convergence. If we compute the error outside a small ball around the origin we find the expected accuracy Op∆s⁴`∆y⁴q.

Figure 4: Coefficients of the average matrix field xI₂y, from left to right and top to bottom, we have xI₂y₁₁,xI₂y₁₂,xI₂y₂₁ andxI₂y₂₂.

10^´2 10^´1

10^´5 10^´4 10^´3 10^´2 10^´1

∆y L2 error

xI₂y

Away from the origin Near the origin

Slop e3.93

Slope 1.13

Figure 5: L² error for the approximation ofxI2y

Two dimensional Fokker-Planck equation

In this example, we approximate the effective diffusion matrix field associated to a model which describe the evolution of a density of charged particles under the action of high magnetic field by taking into account the collision effects. We assume that the magnetic field B^ε is uniform, defined by B^ε“ p0,0, B{εq,B ą0, and that the electric field is given by Ept, xq “ pE₁pt, xq, E₂pt, xq,0qwithtě0 andx“ px₁, x₂q PR². We choose the asymptotic regime with finite Larmor radius that is the typical length in the perpendicular directions (with respect to the magnetic lines) is of the same order as the Larmor radius and the typical length in the parallel direction is much larger. The presence density of a population of charged particles f^ε satisfies the two dimensional Fokker-Planck equation

Btf^ε`1

εpv1Bx1`v2Bx2qf<sup>ε</sup>` q

mE¨∇_vf^ε` qB

mεpv2Bv1´v1Bv2qf^ε“νdivvtΘ∇_vf^ε`vf^εu.

(3.9) Here mis the particle mass,q is the particle charge, ν is the collision frequency and Θ is the temperature. The transport operator associated to the stiff part of the equation (3.9) is

bpx, vq ¨∇_x,v “v1Bx1 `v2Bx2 `ωcpv2Bv1 ´v1Bv2q, with ωc“ qB

m, px, vq PR⁴. The flow of bis denoted

px, vq “ px₁, x₂, v₁, v₂q PR²ˆR²ÞÑYps;x, vq “ pXps;x, vq, Vps;x, vqq PR²ˆR²,

and can be determined explicitly Xps;x, vq “x`

Kv ωc

´Rp´ω_csq ωc

Kv, Vps;x, vq “Rp´ω_csqv

where Rpθq is the two dimensional rotation R² of angle θ P R. We want to compute nu- merically the effective diffusion matrix field associated to the diffusion operator divvp∇_v¨q “ div_x,vpD∇_x,v¨qof the equation (3.9), where the diffusion fieldD is defined by

D“

2

ÿ

i“1

e_vibe_vi “

ˆ 0_2ˆ2 0_2ˆ2 0_2ˆ2 I₂

˙

. (3.10)

For more details on the asymptotic model related to the equation (3.9) see [9]. The Jacobian matrix associated to the flow writes

Bx,vYps;x, vq “

˜

I2 I2´Rp´ωcsq

ωc Rp´π{2q 0_2ˆ2 Rp´ω_csq

¸

where 0_mˆn is the zero matrix with m rows and n columns. The flow Y is ^2π_ω

c-periodic, and a direct computation shows that

xDy px, vq “ lim

SÑ`8

1 S

żS 0

pGpsqDqpx, vqds

“ ω_c 2π

ż_2π{ωc

0

ByYp´s;Yps;x, vqq

ˆ 0_2ˆ2 0_2ˆ2 0_2ˆ2 I₂

˙

tByYp´s;Yps;x, vqqds

“ 1 ω²_c

ˆ 2I2 ´ωcRp´π{2q ω_cRp´π{2q ω_c²I₂

˙

. (3.11)

The method performed in Section 3.2 applies in this situation as well, where the long time average is actually replaced by one period average. However, it is interesting to understand, on this periodic example, the behavior of the numerical error when we compute the long time average. This could be useful when the flow associated to bis periodic whereas the period is not available. In the sequel, we consider that ω_c“1. In order to provide a theoretical study of the error, we introduce some notations. We introduce the set

HQ “

!

A:R⁴ÑM₄pRqmeasurable : Q^1{2AQ^1{2 PL²pR⁴q )

(3.12) whereQ:“ xDy^´1,xDyis a positive definite matrix field such thatxDy PKerL. The setH_Q is a Hilbert space for the natural scalar product

pA, BqH_Q:“

ż

R⁴

pQ^1{2AQ^1{2q:pQ^1{2BQ^1{2qdy“ ż

R⁴

QA:BQdy, @A, BPHQ,

the associated norm is denoted }A}_HQ. It is shown in [3] that the group of linear operators pGpsqq_sPRis aC⁰-group of unitary operators onH_Q. The infinitesimal generatorLassociated to pGpsqqsPR is skew-adjoint in HQ and its kernel coincides with tA P HQ Ă L¹_locpR^mq : rb, As “0PD¹pR^mqu, see [8, 3].

Remark 3.1 Thanks to the Proposition 2.1, the kernel of L can be interpreted as the set of matrix fields A which ensure the commutation property between the operators b¨∇_y and divypA∇_yq.

Moreover, we have the following decompositionHQ“KerL‘^KRangeL. Actually, the uniform boundedness of the flow periods implies the closure of the range of L, see [4]. Thus, for any matrix field APH_Q, there existsC PH_Q such that

A“ xAy `LpCq. (3.13)

Replacing C by C´ xCy, we can also assume thatxCy “0. Recall thatGpsq xAy “ xAy, for any sPR, and therefore, by the unitarity of the group pGpsqqs inHQ, we have

›

›

›

› 1 S

ż_S

0

GpsqAds´ xAy

›

›

›

›HQ

“

›

›

›

› 1 S

ż_S

0

GpsqpLpCqqds

›

›

›

›HQ

“

›

›

›

› 1 S

ż_S

0

d

dsGpsqCds

›

›

›

›HQ

“

›

›

›

›

GpSqC´C S

›

›

›

›HQ

ď 2}C}_HQ S .

Thus, at least in the periodic case, the committed theoretical H_Q error onxAyis O`₁

S

˘. For the numerical simulations, we have to localize the error to a ball Br“ tpx1, x2, v1, v2q PR⁴ : x²₁`x<sup>2</sup><sub>2</sub>`v₁²`v²₂ ď r²u of radius r ą 0. We introduce the notation }A}H_Q,r :“ }1BrA}H_Q

for any matrix field A P L⁸pR^m,M_mpRqq. Thus, we study numerically the error between the quantities S ÞÑ _S¹ş_S

0 GpsqAds and xAy for A “ 1_BrD. In this case, the multiplicative constant in front of the term 1{S can be specified if we can compute the matrix field C and Q. We have

Q“ xDy^´1 “

ˆ I2 Rp´π{2q

´Rp´π{2q 2I₂

˙

(3.14)

100 200 300 400 500 10^´3

10^´2 10^´1 10⁰

Time S L2 error

S ÞÑ }S¹

şS

0 GpsqDds´xDy}_L2

}xDy}_L2

L² error S ÞÑ ^1.6_S

100 200 300 400 500

10^´3 10^´2 10^´1 10⁰

Time S HQ,rerror

SÞÑ }S¹

şS

0GpsqDds´xDy}_HQ,r

}xDy}_HQ,r

HQ,r error S ÞÑ _S²

Figure 6: Long time evolution of the relative error

For the matrix field Ddefined by (3.10), a straightforward computation leads to the matrix field

C “

ˆ02ˆ2 I2

I₂ 0_2ˆ2

˙

(3.15) such that D“ xDy `LpCq and xCy “0. The Figure 6 shows the relative numerical L² and HQ,rerrors committed for the computation of (3.11) by a long time average of the matrix field (3.10). The time resolution is chosen equal to Ns“10³ withS “500. The semi-Lagrangian part of the scheme is realized by a linear interpolation. The volume ofB_ris denoted mespB_rq.

In this case, the expressions (3.11), (3.14) and (3.15) lead to }C}_HQ,r “

d ż

Br

QC : CQdxdv“ d

ż

Br

4 dxdv“2a

mespB_rq and

} xDy }_HQ,r “ dż

Br

QxDy : xDyQdxdv“ dż

Br

4 dxdv“2a

mespB_rq.

Thus ›

›

›

1 S

ş_S

0 GpsqDds´ xDy

›

›

›HQ,r

} xDy }_HQ,r ď 2

S, for anyS ą0.

We retrieved the expected error for the relative H_Q,r error, cf. Figure 6.

Almost periodic flow

We consider the vector fieldbpyq “ py₂,´ω₁²y₁, y₄,´ω²₂y₃q, defined fory“ py₁, y₂, y₃, y₄q PR⁴, withω₁, ω₂PRincommensurable, i.eω₁{ω₂ RQ. The functionψpyq “ω²₁y²₁`y<sub>2</sub><sup>2</sup>`ω²₂y²₃`y₄², withyPR⁴, is a coercive prime integral associated tob. We denoteYps;yqthe flow associated

to the vector fieldb. We consider a constant matrix fieldDPM₄pRq. The flowY is explicit Yps;yq “Rp´s;ω₁, ω₂qy, ps, yq PRˆR⁴, with

Rps;ω₁, ω₂q “

¨

˚

˚

˝

cospsω1q ´_ω¹₁ sinpsω1q 0 0

ω₁sinpsω₁q cospsω₁q 0 0

0 0 cospsω2q ´_ω¹

2 sinpsω2q

0 0 ω₂sinpsω₂q cospsω₂q

˛

‹

‹

‚ .

The incommensurability condition ensures that the flow Yps;¨q is not periodic with respect tos, but almost periodic. For more details on almost periodic functions see [11]. We have

xDy “ lim

SÑ`8

1 S

ż_S

0

BYp´s;Yps;¨qqDpYps;¨qq^tBYp´s;Yps;¨qqds

“ lim

SÑ`8

1 S

żS 0

Rps;ω1, ω2qDpYps;¨qq^tRps;ω1, ω2qds

“

¨

˚

˚

˚

˚

˝

1

2D₁₁`_2ω¹2 1

D₂₂ ¹₂D₁₂´¹₂D₂₁ 0 0

1

2D₂₁´ ¹₂D₁₂ ^ω₂²¹D₁₁`¹₂D₂₂ 0 0

0 0 ¹₂D33`_2ω¹2

2D44 1

2D34´¹₂D43

0 0 ¹₂D₄₃´¹₂D₃₄ ^ω₂²²D₃₃`¹₂D₄₄

˛

‹

‹

‹

‹

‚ .

For this flowY, the ergodic mean can not be reduced to an average over one period as in the periodic case. We need to compute a long time average. We study the error committed with respect to this long time average, when we use the method in Section 3.2, for the matrix field D defined by

D“

¨

˚

˚

˝

2 ´1 0 ´2

´1 1 2 0

0 2 3 1

1 0 ´1 1

˛

‹

‹

‚ .

In the figure 7, we observe that the error isO`₁

S

˘ as in the periodic case. Indeed, the matrix fieldDis constant and the computation of a matrixC which satisfies the equality (3.13) can be provided by solving the linear system LpCq “D´ xDy.

3.4 Shear flow

In this example, we provide a two dimensional vector field such that the associated average matrix field does not exist. Actually, sufficient conditions for the existence of an average matrix field are given in [3]. These conditions are based on the existence of a basis pbiq1ďiď2

of vector fields in involution with the vector field bpyq, i.e when the operators bpyq ¨∇_y and b_ipyq ¨∇_y are commuting, for any 1ďiď2. We define the vector field bpyq “ py₂,´y₁q for y “ py1, y2q P R². In this case, the flow Y associated to b is given by Yps;yq “ Rp´sqy, for any sPR and y PR². Moreover, we consider a smooth even function f :RÑR` such that fp0q “ 1, fpxq “ 0 for any x P r1;`8r and strictly decreasing on r0,1s. Finally, we introduce the radial function Tpyq “fp|y|qfor yPR². We denote by Z the flow associated to the vector field T b. We have Zps;yq “ Rp´sTpyqqy for any s P R and y P R². The flow Z is a rotating shear flow, indeed the rotating period associated at each characteristic is different. If y is a point of the characteristic Zps;yq, for s P R, the associated period is Tpyq. The functionT is constant along the flowsY andZ. We claim that the average matrix field associated toD“I₂, with respect to the vector fieldT bis not well defined, in the sense (2.4), on Bp0,1q{tp0,0qu where Bp0,1q is the unit open ball of R². Indeed, thanks to the

100 200 300 400 500 10^´3

10^´2 10^´1 10⁰

TimeS

ErrorL2

S ÞÑ }S¹

şS

0GpsqDds´xDy}_L2

}xDy}_L2

ErrorL² SÞÑ _S⁴

Figure 7: Long time evolution of the relative error

expression of the group (2.1) associated toT b, for anysPRand yPR², we have

GpsqI2 “ BZ^´1ps;yq^tBZ^´1ps;yq. (3.16) A direct computation shows that, for any sPRand yPR², we have

BZps;yq “ ´sRpπ{2qZps;yq b∇_yTpyq `Rp´sTpyqq.

On the other hand, thanks to the relationBZ^´1ps;yq “ BZp´s;Zps;yqqwhich is available for any sPRand yPR², we obtain

BZ^´1ps;yq “sRpπ{2qyb∇_yTpZps;yqq `RpsTpyqq. (3.17) Moreover, thanks to the equality ∇_yTpyq “ f¹p|y|q

|y| y, we can write (3.17) in the following form

BZ^´1ps;yq “ sf¹p|y|q

|y| Rpπ{2qybZps;yq `RpsTpyqq. (3.18) Finally, we combine (3.16) and (3.18), and a straightforward computation leads to

GpsqI2 “s²f¹p|y|q² pb b bqpyq `s f¹p|y|q

¨

˝

´2y1y2

|y|

y²₁´y²₂

|y|

y²₁´y²₂

|y|

2y1y2

|y|

˛

‚`I2. (3.19) We integrate (3.19) with respect to sover the intervalr0, SswithS ą0, for any yPR², and we obtain

1 S

ż_S

0

pGpsqI₂qpyqds“ S²

3 f¹p|y|q² pb b bqpyq `S

2 f¹p|y|q

¨

˝

´2y1y2

|y|

y²₁´y²₂

|y|

y²₁´y²₂

|y|

2y1y2

|y|

˛

‚`I<sub>2</sub>. (3.20) Thus xI2y p0,0q “I2 and xI2y pyq “I2 for anyy PBp0,1q<sup>c</sup>. But, for yPBp0,1q{tp0,0qu, the expression (3.20) does not have a limit when S Ñ `8. The average matrix field associated toD“I₂ with respect toT bis not well defined.

4 Asymptotic behavior of the solutions of a parabolic problem with stiff transport terms

In this section, we study numerically the asymptotic behavior of the familypu^εqεą0, solutions of (1.1), when εgoes to 0, in the two dimensional settingm“2. The behavior ofu^ε can be described, whenεgoes to 0, as a composition product of a profilev, which does not depend of ε, with the flow associated to the vector field´b{ε. Moreover, the profilevsolves the effective diffusion problem (2.3) with the diffusion matrix field xDy (2.4). Thanks to the method in Section 3 for the computation of the effective diffusion field, we solve the system (2.3), and we study the error between a reference solutionu^εpt,¨qand vpt, Yp´t{ε;¨qq. The anisotropic diffusion equation, with T ą0 andyPR²,

"

Btv´divypxDy∇_yvq “0, pt, yq P r0, Ts ˆR² vp0, yq “uⁱⁿpyq, yPR²

is solved by an implicit method in time and finite difference method in space, see [24]. As in Section 3.2, we introduce a time discretization pt^kq of r0, Ts, with k P I_t “ t0, . . . , Ntu and ∆t “ T{N_t. We consider also a spatial discretization py_ijq, with a step ∆y, for the square C “ r´R, Rs ˆ r´R, Rs, with R ą 0, and we add the points y_{i`1{2,j</sub> and y<sub>i,j`1{2} for any i, j P I “ t0, . . . , N ´1u, for the flux computation. We introduce the notation W“ ´ xDy∇_yv. The operator divypWqat the point yij is approximated by

pdiv_ypWqq_ij «

W¹_i`1{2,j´W¹_i´1{2,j

∆y `

W²_i,j`1{2´W²_i,j´1{2

∆y (4.1)

where

W¹_{i`1{2,j</sub> “ ´pxDy<sub>11</sub>q<sub>i`1{2,j} Bv By1

ˇ ˇ

ˇi`1{2,j´ pxDy<sub>12</sub>q<sub>i`1{2,j</sub> Bv By2

ˇ ˇ ˇi`1{2,j

W²_{i,j`1{2</sub> “ ´pxDy<sub>21</sub>q<sub>i,j`1{2} Bv By1

ˇ ˇ

ˇi,j`1{2´ pxDy<sub>22</sub>q<sub>i,j`1{2</sub> Bv By2

ˇ ˇ ˇi,j`1{2.

(4.2)

The derivatives are approximated by the following expressions Bv

By1

ˇ ˇ

ˇi`1{2,j« vi`1,j´vi,j

∆y Bv

By₂ ˇ ˇ

ˇi,j`1{2« vi,j`1´vi,j

∆y Bv

By₂ ˇ ˇ

ˇi`1{2,j« v<sub>i`1{2,j`1</sub>´v<sub>i`1{2,j´1</sub>

2∆y “ v_i`1,j`1`v<sub>i,j`1</sub>´v_i`1,j´1´v_i,j´1 4∆y

Bv By1

ˇ ˇ

ˇi,j`1{2« v<sub>i`1,j`1{2</sub>´v<sub>i´1,j`1{2</sub>

2∆y “ v_i`1,j`1`v<sub>i`1,j</sub>´v_i´1,j`1´v_i´1,j 4∆y

(4.3)

and the diffusion coefficients xDy_{i`1{2,j</sub>,xDy<sub>i,j`1{2} by

xDy_{i`1{2,j</sub> “ xDy<sub>i`1,j}` xDy_i,j 2

xDy_{i,j`1{2</sub> “ xDy<sub>i,j`1}` xDy_i,j

2 .

(4.4)

We are led to the semi-discrete scheme in space, where we denote byv_ijptqthe approximations of the unknowns vpt, yijqfor any pi, jq PI²

Btvijptq ` W<sup>1</sup><sub>i`1{2,j</sub>ptq ´W¹_i´1{2,jptq

∆y `

W²_i,j`1{2ptq ´W²_i,j´1{2ptq

∆y “0.