A sixth-order finite volume method for multidomain convection–diffusion problem with discontinuous coefficients

Comput. Methods Appl. Mech. Engrg. 267 (2013) 43–64

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Comput. Methods Appl. Mech. Engrg.

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A sixth-order finite volume method for multidomainconvection–diffusion problem with discontinuous coefficients

0045-7825/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cma.2013.08.003

⇑ Corresponding author at: Centre of Mathematics, University of Minho, Campus de Azurém, 4080-058 Guimarães, Portugal. Tel.: +351 96960E-mail addresses: [email protected] (S. Clain), [email protected] (G.J. Machado), [email protected] (J.M. Nóbrega), rmp@

minho.pt (R.M.S. Pereira).

S. Clain a,c,⇑, G.J. Machado a, J.M. Nóbrega b, R.M.S. Pereira a

a Centre of Mathematics, University of Minho, Campus de Azurém, 4080-058 Guimarães, Portugalb Institute for Polymers and Composites/I3N, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugalc Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 April 2013Received in revised form 31 July 2013Accepted 5 August 2013Available online 25 August 2013

Keywords:Finite volumeHigh-orderConvection–diffusionPolynomial reconstructionHeat transferDiscontinuous coefficients

A sixth-order finite volume method is proposed to solve the bidimensional linear steady-state convection–diffusion equation. A new class of polynomial reconstructions is proposedto provide accurate fluxes for the convective and the diffusive operators. The method is alsodesigned to compute accurate approximations even with discontinuous diffusion coeffi-cient or velocity and remains robust for large Peclet numbers. Discontinuous solutionsderiving from the linear heat transfer Newton law are also considered where a decompo-sition domain technique is applied to maintain an effective sixth-order approximation.Numerical tests covering a large panel of situations are addressed to assess the perfor-mances of the method.

� 2013 Elsevier B.V. All rights reserved.

1. Finite volume scheme for the convection–diffusion problem

1.1. Introduction

Finite volume method for linear convection–diffusion problems has received a great interest since the eighties, initiatedwith the well-known book of Patankar [1]. Second-order schemes based on linear reconstruction, coupled with a MUSCLtechnique to guarantee the stability of the convection contribution are very popular and have been extensively studied bothfor structured and unstructured meshes. Since this date, important developments for second-order approximations havebeen realized and several classes of methods have been proposed. First, the original Patankar scheme for structured mesheshas been extended to the non-structured case where an orthogonality condition is required to allow admissible diffusion flux(FV4 scheme [2,3,5,6]). The diamond scheme based on a local reconstruction of the gradient on each edge has been intro-duced in [7–9] while a finite volume scheme based on primal and dual meshes (DDFV scheme) has been proposed and devel-oped in [10–12]. New techniques to design efficient finite volume schemes have been recently developed and a largeproposal of numerical algorithms is now available such as the mixed-hybrid schemes [13,14] or the mimetic schemes[15,16].

In the last three decades, very high-order methods based on polynomial reconstruction and a WENO procedure have beenproposed to achieve third-order, and up to fifth-order scheme [17–21]. The main ingredient is a k-exact polynomial recon-

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44 S. Clain et al. / Comput. Methods Appl. Mech. Engrg. 267 (2013) 43–64

struction introduced by Barth and Frederickson [22,23]. Numerical fluxes are evaluated with the help of the local reconstruc-tions while a nonlinear procedure is employed to reduce the method order in the vicinity of discontinuities and to avoid theappearance of spurious oscillations. The recent MOOD method developed in [24–26] allows an a posterior control of thereconstruction degree in function of the local regularity and provides accurate approximations for passive convection prob-lem or the Euler system.

While polynomial reconstructions applied to hyperbolic problems has been widely investigated, its application to theelliptic operator context did not receive so much attention and only recent studies have emerged following the pioneer arti-cle of Ollivier-Gooch and Van Altena [18]. Very high-order finite volume method for the elliptic operator was, at first, mainlymotivated by the compressible Navier–Stokes equations where the traditional Euler system is augmented with an accuracyapproximation of the viscous term to provide a global scheme of the same order [20,27]. Other class of problems such as thepopular convection–diffusion problem with large Peclet number also requires an accurate resolution both for the hyperbolicand the elliptic part of the operator and the finite volume method turns out to be a good candidate to provide the requiredaccuracy and robustness.

In the present article, we propose a sixth-order finite volume method for the two-dimensional linear convection–diffu-sion problem based on polynomial reconstructions. It is important to notice that we did not deliberately introduce any lim-itation procedure since our goal is to provide very high approximations assuming the solution is smooth enough. Of course,dealing with a nonlinear problem will require some limitation procedure since discontinuities may appear but a very high-order scheme far from the discontinuity regions is a fundamental building-block. Even for smooth solutions, specific diffi-culties for elliptic operators arise such as the odd–even solution decoupling problem [28] or the non-negative entries ofthe inverse matrix which guarantees the positivity preserving property.

A first issue of the paper is to tackle the stability question and to propose a very-high order method which provides aglobal underlying matrix A which the inverse has non-negative entries. The second issue we address is to design a sixth-or-der method where the diffusion or the velocity may comprise some discontinuities. At last, we intend to build a matrix-freemethod where no explicitly linear system is stored. To this end, an affine operator U! GðUÞ is defined such that the zeroprovides the numerical approximation. Such a point would be of crucial importance from a computational point of viewwhen one has to deal with a large number of cells.

The paper is organized as follows. After a presentation of the generic convection–diffusion problem in Section 1, we intro-duce the polynomial reconstruction in the second Section of the document. Section 3 is devoted to the finite volume scheme.Numerical tests are given in the fourth Section and the study ends with a conclusion and some perspectives.

1.2. The steady-state problems

Let X be a bounded polygonal domain of R2 and @X its boundary. Since we shall consider situations where the diffusionand the convection functions may comprise discontinuities, domain X is partitioned in two subdomains X1 and X2 sharing acommon interface C (see Fig. 1, left panel). We denote by k1 and k2 positive regular functions on X1 and X2 respectively,while V1 ¼ ðu1;v2Þ and V2 ¼ ðu2;v2Þ stand for the velocities. For the sake of simplicity we also use the notation k wherekjX1 ¼ k1 on X1 and kjX2 ¼ k2 on X2 (the same for V). To preserve the mass flow across the interface, we prescribe thatV1:nC ¼ V2:nC ¼ 0, with nC the outward normal vector of X1 on C. We also consider a partition of the boundary @X into threeparts, namely CD, CP , and CT corresponding to the Dirichlet condition, the Neumann condition for the diffusive flux, and theNeumann condition for the total flux respectively.

When dealing with two different domains, a first approach consists in prescribing the continuity of function / across theboundary and the problem Pc writes: find /1 and /2 such that

r:ðV1/1 � k1r/1Þ ¼ f1; in X1; ð1aÞ

r � ðV2/2 � k2r/2Þ ¼ f2; in X2; ð1bÞ

k1r/1 � nC ¼ k2r/2 � nC; on C; ð1cÞ

/1 ¼ /2 on C; ð1dÞ

Fig. 1. Domain parted by interface C (left). Mesh and notations (right).

S. Clain et al. / Comput. Methods Appl. Mech. Engrg. 267 (2013) 43–64 45

/ ¼ /D; on CD; ð1eÞ

�kr/ � n ¼ gP; on CP; ð1fÞ

V � n/� kr/ � n ¼ gT ; on CT ; ð1gÞ

where /D; gP , and gT are prescribed functions for the Dirichlet condition, the Partial flux, and the Total flux respectively, fi

stands for the source term in each subdomain i ¼ 1;2, and n represents the outward normal vector on the boundary.The second problem consists in substituting the continuity condition (1d) at the interface with a flux transfer condition

leading to problem Pf : find /1 and /2 such that Eqs. (1a)–(1c) and (1e)–(1g) hold and we prescribe on interface C

k1r/1 � nC ¼ hð/2 � /1Þ; ð2Þ

with h a positive function defined on C. We point out that, in the last case, the solution is not necessarily continuous acrossthe interface. Moreover, when h!1, we shall recover the continuous case and problem Pf can be seen has a penalizationmethod of problem Pc.

1.3. Mesh and discretization

We consider a partition T of X into non-overlapping convex polygonal cells ci2C (I ¼ #C the number of cells) whichmatches with the interface C. To design the very high-order finite volume method, we adopt the following conventions(see Fig. 1, right panel).

� Inner edge: two cells ci and cj share a common edge eij and nij is the normal vector from ci toward cj, i.e. nij ¼ �nji.� Edge on the boundary @X: a cell ci shares a common edge with the boundary that we shall denote eiD (resp. eiP or eiT ) if the

edge belongs to CD (resp. CP or CT ). Note that the convention implies that niD (resp. niP or niT ) corresponds to the outwardnormal vector.� Edge on interface C: the situation is very similar to the first one but we adopt the convention that ci belongs to domain X1

while cell cj belongs to domain X2 and one has nij ¼ nC.� To deal with the edges on the boundary, we augment the cell index C with the indexes D; P and T and setCb ¼ C [ fD; P; Tg.� For any cell ci; i 2 C, we associate the index set mðiÞ � Cb such that j 2 mðiÞ if eij is a common edge between ci and cj or with

the boundary if j ¼ D; P; T .� We denote by jcij and jeijj; i 2 C; j 2 mðiÞ, the area and length of ci and eij respectively.� For any edge eij; i 2 C; j 2 mðiÞ, we denote by qij;r the Gauss points with weights fr ; r ¼ 1; . . . ;R, while mij represents the

midpoint of the edges. Note that miD; miP; miT correspond to the midpoints of edges lying on CD; CP ; CT respectively.

Integrating Eqs. (1a) and (1b) over cell ci and applying the divergence theorem yields,
Z@ci
ðV � n/� kr/ � nÞds�Z

ci

f dx ¼ 0:

We first determine an accurate approximation fi of the mean value over cell ci using a quadrature rule on triangles asproposed in [4]. When dealing with a polygonal cell, we split the cell into sub-triangles which share the cell centroid as acommon vertex and apply the quadrature rule for each sub-triangle. For example, we use a 5 Gauss points and 15 Gausspoints to provide an exact quadrature rule on triangle for polynomials P3 and P5 respectively. Using the Gauss quadratureformula with R points on the edges provides the approximation
X
j2mðiÞjeijjXR

r¼1

fr Vðqij;rÞ � nðqij;rÞ/ðqij;rÞ � kðqij;rÞr/ðqij;rÞ � nðqij;rÞ� �

� jcijfi ¼ O ðh2Ri Þ;

where fi is an accurate approximation of the mean value of f over ci, exact for P2R�1 polynomial functions whilehi ¼maxk2mðiÞjeikj.

Based on the previous expression, we consider a numerical scheme under the residual form,

Gi ¼Xj2mðiÞ

jeijjjcijXR

r¼1

frF ij;r � fi;

where F ij;r stands for the numerical flux approximation of the physical flux function evaluated at the Gauss point qij;r .

2. The Polynomial Reconstruction Operator

To provide accurate approximations of the flux across the edges, Polynomial Reconstruction Operators (PRO) are intro-duced in order to recover local polynomial functions associated to cells or edges. Initially introduced in [22,23] for hyperbolic


problems, extensions have been proposed in the elliptic context for diffusive or viscous terms in [18,20,27]. The reconstruc-tion procedure is a powerful tool to achieve accurate representative functions when dealing with piecewise constant approx-imations. For that purpose, we propose here some modifications of the original definition, in order to better approximate thediffusive flux and to integrate the boundary conditions. We introduce different types of reconstructions, namely the conser-vative and the non-conservative ones, since reconstruction associated to an edge is basically dedicated to the diffusion fluxcomputation, whereas polynomial function associated to a cell is employed for the convection flux calculation.

2.1. Stencil

Roughly speaking, a stencil is a collection of cells in the vicinity of a reference geometrical entity (edge or cell). We shallconsider two situations whether we deal with a cell or an edge. For a given cell ci and a polynomial degree d, we associate thestencil Sðci; dÞwhile Sðeij; dÞ; j 2 mðiÞ, stands for the stencil associated to edge eij. We would like to highlight the importance ofcomposing a stencil with cells of the same domain. For instance, if ci � X1, then all the cells of Sðci; dÞ belong to X1. We applythe same strategy for the inner edges eij of each subdomain whereas no stencil will be considered for eij � C since we shallnot use any reconstruction associated to edges on the interface C. It should be noticed that this strategy is a key-point toprovide very high-order approximation even for discontinuous coefficients. In some way, the method is similar to a domaindecomposition technique since we perform the reconstruction with the data of each subdomain.

To provide a relevant stencil, we choose the Nd cells belonging to the same domain closest to ci or to eij. Since the poly-nomial of degree d has ðdþ 1Þðdþ 2Þ=2 coefficients, we take Nd P ðdþ 1Þðdþ 2Þ=2. In practice we chooseNd 2�ðdþ 1Þðdþ 2Þ=2; ðdþ 1Þðdþ 2Þ� for the sake of robustness.

2.2. Conservative reconstruction for cells

Let ci be a cell, d the degree of the polynomial reconstruction we intend to build, and Sðci; dÞ the associated stencil. Weassume that mean value /i on ci is given and we consider the polynomial function of degree d, cast in the following form
b/iðx; dÞ ¼ /i þ
X16jaj6d

Rd;ai ðx� biÞa �Ma

i

� �; ð3Þ

with a ¼ ða1;a2Þ; jaj ¼ a1 þ a2; x ¼ ðx1; x2Þ; bi the centroid of cell ci, and we adopt the convention xa ¼ xa11 xa2

2 . We setMa

i ¼ 1jci jR

ciðx� biÞa dx such that the conservation property

1jcij

Zci

b/iðx; dÞ dx ¼ /i ð4Þ

holds. This polynomial expression contains explicitly the conservation property whereas the traditional method introducesthe restriction via the least-square procedure solving the over-determined system [18,20,22]. It results that our approachstrictly satisfies relation (4). To fix the coefficients Rd;a

i of polynomial function (3) we assume that values /‘ on cellsc‘; ‘ 2 Sðci; dÞ are known, and introduce the functional

EiðRdi ; dÞ ¼

X‘2Sðci ;dÞ

1jc‘j

Zc‘

b/iðx; dÞ dx� /‘

� �2

; ð5Þ

where Rdi is the vector gathering all the coefficients Rd;a

i . Existence and uniqueness of the functional minimizer bRdi provide

the best approximation.

Remark 2.1. Weight coefficients with respect to the centroid–centroid distance are usually applied [20,27] in theminimizing functional to reinforce the reference cell mean value contribution. We do not use weights for the conservativereconstruction (each cell has the same unit weight) since the reference cell mean value is contained in the specificpolynomial expression. Nevertheless, as we shall see in the sequel, weight coefficients will be of interest but for a radicallydifferent reason, namely the M-matrix property in the elliptic context.

The minimizing point of functional (5) corresponds to the solution (in the least-square sense) of an over-determined lin-ear system Ad

i Rdi ¼ bS

i where bSi represents the variations /‘ � /i; ‘ 2 Sðci; dÞ, while matrix Ad

i only depends on the mesh andcan be evaluated during a pre-processing step. To reduce the condition number of the system (see [29]), we determine theMoore–Penrose pseudo-inverse matrix for system ðAd

i Pdi ÞðPd

i Þ�1

Rdi ¼ bd

i where Pdi is the diagonal matrix

Pdi ¼ diagðjcij�jaj=2Þ16jaj6d:

Indeed, Pdi diagonal matrix coefficient jcij�jaj=2 is multiplied with the Ad

i matrix coefficient

1jc‘j

Zc‘

ðx� biÞadx� 1jcij

Zci

ðx� biÞa dx


which strongly reduces the effect of the power a (smaller coefficients for larger degrees). Let us denote by ðAdi P

di Þy

the pseudoinverse matrix. Hence one has

Rdi ¼ Pd

i ðAdi P

di Þybd

i :

Since Adi and Pd

i do not depend on the solution, the pseudo-inverse matrix is also evaluated during the pre-processingstage.

2.3. Conservative reconstruction for cells in contact with C

To compute an accurate diffusion flux across interface C when dealing with the continuous transition, we introduce a spe-cific reconstruction for the cells cj � X2 which share an edge eij � C with the interface. As in the previous case, we consider apolynomial function under its conservative form

/jðx; dÞ ¼ /ij þX

16jaj6d

Rd;aj ðx� bjÞa �Ma

j

n o:

We now assume that there exists a relevant mean value approximation /ij on edge eij and we slightly modify the functionto minimize, namely

EjðRdj ; dÞ ¼

X‘2Sðcj ;dÞ

1jc‘j

Zc‘

/jðx; dÞ dx� /‘

� �2

þxij1jeijj

Zeij

/jðx; dÞ ds� /ij

" #2

ð6Þ

with xij a positive weight. The functional takes now into account the continuity condition of the reconstruction across Ccontrolled by parameter xij to be fixed (when xij increases, the continuity condition is reinforced and in the numerical testswe consider xij ¼ 3). The minimizing vector Rd

j is solution of an over-determined system, in the least-square sense, we havepreconditioned as in the previous case.

2.4. Non-conservative reconstruction for edges

We turn now to the polynomial reconstruction associated to an edge. A common usage to compute the diffusion fluxacross edge eij consists in using a convex combination of the two conservative reconstructions on both sides of the edge[18,20], namely taking for each Gauss point

kðqij;rÞ12r b/iðqij;rÞ þ b/jðqij;rÞh i

� nðqij;rÞ:

Such a technique may suffer of two major drawbacks: an odd–even solution decoupling phenomenon and the matrixassociated to the diffusion problem could not be an M-matrix.

Let us consider the simple one-dimensional pure diffusion problem with a uniform mesh. The P1 reconstruction (linearreconstruction) on cells generates an odd–even solution decoupling where the matrix is composed of two independent sub-systems coupled only by the boundary conditions. A reconstruction based on edges avoids such a problem since the P1 poly-nomial reconstruction (Patankar flux in the 2D or 3D structured meshes) provides an M-matrix. To deal with accurateapproximation of the diffusive flux across the edge, we introduce a non-conservative reconstruction since no mean valueis associated to the edge (except for the Dirichlet condition) and we use the neighbor cell mean values to fix the polynomialcoefficients. To this end, we consider the polynomial reconstruction of the form
e/ijðx; dÞ ¼
X06jaj6d

Rd;aij ðx�mijÞa; ð7Þ

where mij is the midpoint of edge eij. Notice that the index summation starts with jaj ¼ 0. Assuming that we know /‘ on cellsc‘; ‘ 2 Sðeij; dÞ, we introduce the functional

EijðRdij; dÞ ¼

X‘2Sðeij ;dÞ

xij;‘1jc‘j

Zc‘

e/ijðx; dÞ dx� /‘

� �2

; ð8Þ

where xij;‘ are positive weights and Rdij is the vector gathering all the coefficients Rd;a

ij . Weights are introduced to provide thepositivity preserving property and are defined in the following way: if ‘ ¼ i; j; xij;‘ ¼ x P 1 else xij;‘ ¼ 1. The choice of sucha penalization is justified by the following argument: we reinforce the adjacent cells influence. For large values of x, theresulting matrix is closer to the one deriving from Section 3.3 which is a global M-matrix. As we shall see in the numericalsection, such a technique effectively provides an M-matrix for x large enough (x ¼ 3 in our applications) but with a slightincrease of the condition number of the global matrix.

Vector eRdij stands for the unique vector minimizing the functional and provides the best approximation. As in the conser-

vative case, we use the preconditioning technique involving the diagonal matrix Pdi and we pre-compute the pseudo-inverse

matrix.


2.5. Conservative reconstruction for edges with the Dirichlet condition

We now have to consider a specific case when eiD is an edge situated on the boundary CD. Indeed the mean value/iD ¼ 1

jeiD jR

eiD/DðsÞ ds is relevant and the polynomial reconstruction on edge eiD turns to be a conservative one considering

the polynomial function of degree d
b/iDðx; dÞ ¼ /iD þX
16jaj6d

Rd;aiD ðx�miDÞa �Ma

iD

� �; ð9Þ

with miD the midpoint of edge eiD. We set MaiD ¼ 1

jeiD jR

eiDðx�miDÞa ds such that the conservative property

1jeiD jR

eiD

b/iDðx; dÞ ds ¼ /iD holds. Note that the Dirichlet boundary condition for the diffusive contribution is automatically ta-ken into account thanks to this specific reconstruction.

Let /‘ be the values on cells c‘; ‘ 2 SðeiD; dÞ. As in the previous case, we introduce the functional

EiDðRdiD; dÞ ¼

X‘2SðeiD ;dÞ

xiD;‘1jc‘j

Zc‘

b/iDðx; dÞ dx� /‘

� �2

; ð10Þ

where xiD;‘ are positive weights and RdiD is the vector gathering all the coefficients Rd;a

iD . Weights are introduced to providethe M-matrix property and are defined in the following way: if ‘ ¼ i; xiD;‘ ¼ x P 1 else xiD;‘ ¼ 1. Vector bRd

iD stands for theunique vector minimizing the functional and providing the best approximation.

2.6. Linearity and consistency of the reconstruction operators

We have defined four polynomial reconstructions b/i; /j;b/iD, and e/ij which correspond to linear operators with respect

to the unknown mean values vector U ¼ ð/iÞi2C (the mean values of /D on CD are known thanks to the Dirichlet condition).We say that the reconstructions are d-consistent (or d-exact, or invariant in Pd) in the following sense: for any polynomial

function / of degree d, let denote by /i and /iD the mean values on the cells and edges of CD and C, the reconstruction is d-consistent if taking /i and /iD to perform the reconstruction process one has:
b/iðx; dÞ ¼ /jðx; dÞ ¼ b/iDðx; dÞ ¼ e/ijðx; dÞ ¼ /ðxÞ; x 2 R2:
If such a property holds, we say that the finite volume method associated to the polynomial reconstruction (in short PRO-d) is a dþ 1th-order method.

3. High-order finite volume scheme

3.1. Numerical scheme for problem Pf

Based on the polynomial reconstructions, we now design the finite volume scheme when dealing with the interface heatflux conditioned by the temperature differences on both sides of the interface. We assume that d is fixed and that the stencilswe employ, guarantee the d-consistency of the reconstruction. We introduce the numerical flux in function across the edgeswhere we distinguish five situations listed below (with notation ½v �þ ¼maxð0;vÞ and ½v �� ¼minð0;vÞ).

I. Assume that eij is an inner edge, then we define the flux at the quadrature point by

F ij;r ¼ ½Vðqij;rÞ:nij�þb/iðqij;r; dÞ þ ½Vðqij;rÞ:nij��b/jðqij;r ; dÞ � kðqij;rÞre/ijðqij;r ; dÞ � nij:

II. Assume that eij is on interface C, then we define the flux at the quadrature point by

F ij;r ¼ hðqij;rÞ½b/iðqij;r ; dÞ � b/jðqij;r; dÞ�

with h the heat transfer coefficient given in (2)III. Assume that eiD belongs to CD, then the flux at the quadrature point writes

F iD;r ¼ ½VðqiD;rÞ � niD�þb/iðqiD;r ; dÞ þ ½VðqiD;rÞ � niD��/DðqiD;rÞ � kðqiD;rÞrb/iDðqiD;r; dÞ � niD:

IV. Assume that eiP is on CP , then the flux at the quadrature point writes

F iP;r ¼ VðqiP;rÞ � niPb/iðqiP;r ; dÞ þ gPðqiP;rÞ:

V. Assume that eiT is on CT , then the flux at the quadrature point writes

F iT;r ¼ gTðqiT;rÞ:

For any vector U 2 RI , we define the residual on cell ci by


GiðUÞ ¼Xj2mðiÞ

jeijjjcijXR

r¼1

frF ij;rðUÞ � fi ð11Þ

and introduce the linear operator GðUÞ ¼ ðG1ðUÞ; . . . ;GIðUÞÞT which corresponds to an affine operator from RI into RI . Thesolution is vector bU such that GðbUÞ ¼ 0 while the consistency error is given by GðUÞ with /i the mean values of the exactsolution.

3.2. Numerical scheme for problem Pc

We now turn to the case where heat transfer is imposed by the temperature continuity (equivalent to h ¼ þ1). To guar-antee the continuity across the interface, we proceed in three steps.

We first determine all the polynomial reconstructions for cells belonging to domain X1, in particular, we compute b/i forthe cells in X1 which share a common edge eij with C.

Secondly, for edges eij � C, we evaluate

/ij ¼XR

r¼1

nrb/iðqij;r; dÞ;

which provides the mean value of b/i over eij.At last, with the help of /ij, we compute /jðx; dÞ on the other side of the interface for cell cj � X2, and we complete the

reconstruction process computing the other polynomial functions.For this case, we have five situations similar to the previous case where only the flux on the interface is different. Con-

sequently point (II) in paragraph 3.1 is replaced by

F ij;r ¼ �k2ðqij;rÞr/jðqij;r; dÞ � nji:

We define the residual on cell ci for any vector U 2 RI by

GiðUÞ ¼Xj2mðiÞ

jeijjjcijXR

r¼1

frF ij;rðUÞ � fi ð12Þ

and we set GðUÞ ¼ ðG1ðUÞ; . . . ;GIðUÞÞT . We obtain an affine operator from RI into RI with which the solution is the vector bUsuch that GðbUÞ ¼ 0 while the consistency error is given by GðUÞ with /i the mean values of the exact solution.

3.3. Preconditioning the matrix-free problem

The residual form provides a linear operator U! GðUÞ which corresponds to an underlying matrix A such thatGðUÞ ¼ AU� b. Since we shall use a matrix-free method, the GMRES solver is employed where we compute a sequence ofvectors Uk (U0 is given) and the associated Krylov spaces R0;AR0

; . . . ;AkR0 with R0 ¼ AU0 � b ¼ GðU0Þ (an orthogonalizationof the Krylov basis is also performed). In the context of operator G, vectors AkR0 are obtained iteratively withAkR0 ¼ GðAk�1R0Þ þ Gð0Þ (of course we compute b ¼ �Gð0Þ on the preprocessing stage). When the residual norm saturates,we apply the restart technique taking U0 equal to the former last iteration approximation.

Preconditioning operator G is a key-point to dramatically reduce the computational effort. Classical techniques are avail-able such as the incomplete LU decomposition (see for instance [30,31]) usually employed for sparse matrices. However, inthe present situation, we do not have explicitly the matrix and such a strategy does not apply any longer. The idea is theintroduction of a simple approximated matrix of A we shall use as a candidate matrix to design a preconditioning matrix.We first consider the simple diagonal matrix DP which corresponds to the diagonal matrix deriving from a Patankar-like dis-cretization including both the diffusion and the convection contributions

DPði; iÞ ¼1jcijXj2mðiÞjeijj

kðmijÞjbibjj

þ ½VðmijÞ:nij�þ� �

;

where jbibjj is the distance between the two centroids. When dealing with an edge on the boundary associated to the Dirich-let condition, we substitute jbibjj by jbimijj. Notice that the contribution of the convection part is of crucial importance whendealing with large Péclet numbers or pure convection problem. The inverse matrix is easily evaluated and preconditioning isachieved applying the GMRES algorithm to operator U! D�1

P GðUÞ which dramatically reduces the number of iterations.To provide more sophisticated preconditioning matrix, we use the full Patankar-like matrix AP as an approximation of

matrix A given by APði; iÞ ¼ DPði; iÞ for the diagonal coefficients and

APði; jÞ ¼jeijjjcij

� kðmijÞjbibjj

þ ½VðmijÞ � nij��

: j 2 mðiÞ:


Notice that AP is a diagonal dominant Z-matrix by construction and provide an inverse matrix with non-negative entries.The preconditioned matrix is supposed to be the inverse matrix A�1

P but we shall substitute it with an incomplete inverse AyPwith the non-null entries AyPði; iÞ, AyPði; jÞ; j 2 mðiÞ for the sake of simplicity. We do not perform an incomplete LU decompo-sition as usual since the structure of AP and AyP allows an explicit construction of AyP . Indeed, we seek the AyP coefficients suchthat

AyPAP ¼non-null entries Id;

where the identification is only performed with the non-null entries.The specific structure of the matrices provides the following relations

AyPði; iÞAPði; iÞ þXj2mðiÞ

AyPði; jÞAPðj; iÞ ¼ 1;

AyPði; iÞAPði; jÞ þ AyPði; jÞAPðj; jÞ ¼ 0; j 2 mðiÞ:

We then deduce

AyPði; jÞ ¼ �APði; jÞAyPði; iÞAPðj; jÞ

; j 2 mðiÞ

with

AyPði; iÞ ¼1

APði; iÞ �P

j2mðiÞAP ði;jÞAP ðj;iÞ

AP ðj;jÞ

:

The GMRES method is then performed with the preconditioning operator U! AyPGðUÞ. We assess the preconditioningtechnique performance in the numerical section.

4. Numerical tests

The present section is dedicated to the quantitative and qualitative assessments of the developed schemes robustness andaccuracy. We first deal with smooth solutions for pure convection, pure diffusion, and convection–diffusion problems withlow and large Péclet number, to evaluate the convergence rate of the method. Then we investigate the heat transfer problemwith discontinuous diffusion coefficients considering the two models (linear heat transfer or continuity across an interface)where high-order effective convergences are reported. At last we consider a typical application involving heat transfer inpolymer cooling procedure and highlight the efficiency of the preconditioning matrix.

To measure the accuracy, two error estimators are employed namely

E1 ¼XI

i¼1

jcijj/i � /ij; E1 ¼ maxi¼1;...;I

j/i � /ij:

The first norm gives a global evaluation of the error while the second norm provides the highest local error. The conver-gence order is given by ord ¼ 2 ln errðN1Þ

errðN2Þ

� = ln N2

N1

�where errðN1Þ (resp. errðN2Þ) corresponds to the error computed with a

mesh of N1 (resp. N2) cells. Notice that factor 2 derives from the space dimension.

4.1. Inverse matrix non-negativity

As a first issue, we check that the underlying global matrix A deriving from operator GðUÞ ¼ AU� b has a non-negativeinverse matrix when dealing with a pure diffusive problem, namely all the entries of A�1 are non-negative. Such propertyis fundamental to guarantee the local maximum principle with null source term i.e. f ¼ 0,

/ðxÞ 2 min@X

/D;max@X

/D

� �
and the positivity preservation
if/D P 0 and f P 0 then / P 0:

For the sake of simplicity, we shall deal with a pure diffusion problem where function /ðx; yÞ ¼ xð1� xÞ þ yð1� yÞ is thesolution on the unit square (hence /D ¼ 0 on @X) of the Poisson equation with homogeneous Dirichlet condition and thesource term writes f ðx; yÞ ¼ 4 P 0.

To check the property, one has to determine explicitly the matrix. We first compute the right hand-side term settingb ¼ �Gð0Þ, then we compute successively the matrix columns Ai ¼ GðeiÞ þ b where ei ¼ ðdijÞj¼1;...;I is a generic vector of thecanonical basis.

Numerical simulations are carried out with a P1 reconstruction since a larger degree will provide the exact solution anddoes not reveal the numerical difficulties we intend to highlight. The polynomial reconstruction depends on the weights


introduced in functionals (8) and (10) since only reconstructions on edges are required. A first attempt consists of using auniform distribution setting xij;‘ ¼ 1; ‘ 2 Sðeij; dÞ, i.e. x ¼ 1. We determine A and compute A�1 with the Gauss eliminationmethod. We observe that A�1 contains 46% of negative values which clearly indicates that A�1 is not a non-negative matrix.Moreover, Fig. 2 top, left panel shows the numerical approximation where the oblique stripped cells correspond to the oneswhich do not respect the positivity principle while the horizontal stripped cells correspond to the ones which are larger to 1

4,the maximum of the exact solution.

To overcome the problem, we increase the weights for the cells adjacent to the edge setting xij;‘ ¼ 2:5 for ‘ ¼ i; j, i.e.x ¼ 2:5, for only the two cells in contact with eij. We now obtain 25% of negative values for A�1 and Fig. 2 top, right panelshows the numerical approximation. We observe there is no more magenta cell which violate the positivity principle and thenumber of white cells is reduced.

In the last attempt, we set xij;‘ ¼ 3, i.e. x ¼ 3, for the two adjacent cells ‘ ¼ i; j while the other weights remain equal toone. We check that matrix A�1 has non-negative entries and the approximation suits very well with the exact solution asshown in Fig. 2 bottom left (approximation) bottom right (exact solution).

Numerical experiences have first been carried out with a linear reconstruction but was extended to higher polynomialreconstruction till degree 5. In all the cases, the property is achieved if one choose a weight x larger than 3 for the adjacentcells.

Fig. 2. Maximum principle property for different weights x: all the weights are equal to 1 (top left), the adjacent cells has a weight equal to 2:5 (top right),the adjacent cells has a weight equal to 3 while the right panel displays the exact solution (bottom left). The exact solution is plotted in the bottom rightpanel for comparison.


Remark 4.1. In the following, all the computations will be performed with a weight of 3 for the adjacent cells and a weightof 1 for the other cells. The authors would like to point out that the choice of the weight x is crucial to provide correctnumerical approximations not only when dealing with a diffusion problem but also for general convection–diffusionproblems.

Remark 4.2. We have also tested if matrix A is an M-matrix but all the numerical experiments we have performed show thatthere always exist positive extra-diagonal entries (we do not have a Z-matrix). The percentage of positive extra-diagonalentries is low for the P1 reconstruction (around 6%) while we have measured about 30% of positive entries for P5 polynomialreconstructions.

4.2. Convection–diffusion problem with smooth coefficients

Convection–diffusion problem on domain X ¼ 0;1� ½2 is solved using the finite volume method equipped with the polyno-mial reconstructions. We check the scheme ability to compute accurate approximations for low and large Péclet numbersstill providing an oscillation-free numerical solution. We assume a normalized diffusion coefficient (we take k ¼ 1) and aconstant velocity V ¼ ðu;vÞ where two situations will be investigated: a small Péclet number (diffusive regime) takingV ¼ ð1;2Þ and a large Péclet number (convective regime) with V ¼ ð102;102Þ.

To perform the numerical tests and compare with an exact solution, we introduce functions

aðxÞ ¼ 1u

x� eux � 1eu � 1

� ; bðyÞ ¼ 1

v y� evy � 1ev � 1

�

Fig. 3. The mesh for low Péclet number (left) and large Péclet number (right).

Fig. 4. Numerical approximation for low Péclet number (left) and large Péclet number (right).

Table 1L1 and L1 errors and convergence rates for low Péclet number.

Nb of cells P1 P3 P5

err1 err1 err1 err1 err1 err1

290 1.04e�02 . . . 4.79e�02 . . . 2.60e�04 . . . 1.52e�03 . . . 8.24e�06 . . . 3.49e�05 . . .

1042 3.02e�03 1.8 1.73e�02 1.6 1.75e�05 4.2 1.52e�04 3.6 1.01e�07 6.8 1.28e�06 5.24256 8.98e�04 1.7 5.33e�03 1.7 1.25e�06 3.8 1.54e�05 3.3 1.43e�09 6.0 1.55e�08 6.3

16,858 4.00e�04 1.2 4.01e�03 0.5 1.02e�07 3.7 1.09e�06 3.8 3.20e�11 5.5 4.01e�10 5.3

Table 2L1 and L1 errors and convergence rates for large Péclet number. The last line (1 vs 4) assesses the convergence error using the first and the last line of the table.



9196 7.27e�04 . . . 1.36e�02 . . . 3.34e�05 . . . 1.14e�03 . . . 2.19e�05 . . . 8.57e�04 . . .

15,486 4.59e�04 1.8 9.24e�03 1.5 1.09e�05 4.3 5.40e�04 2.9 6.28e�06 4.8 1.62e�04 6.422,380 3.10e�04 2.1 8.47e�03 0.5 6.94e�06 2.6 3.16e�04 2.9 1.87e�06 6.6 5.21e�05 6.133,532 2.16e�04 1.8 6.13e�03 1.6 3.62e�06 3.2 1.47e�04 3.8 4.94e�07 6.5 1.37e�05 6.61 vs 4 1.9 1.2 3.4 3.2 5.9 6.4


and one can verify that function /ðx; yÞ ¼ CaðxÞbðyÞ;C 2 R, is the solution of problem r � ðV/�r/Þ ¼ f withf ¼ C½aðxÞ þ bðyÞ� as the source term and prescribing homogeneous Dirichlet condition. We adapt constant C to provide anormalized solution such that the maximum is equal to one: C ¼ 11236 for the large Péclet number and C ¼ 65 for thelow Péclet number case.

We plot in Fig. 3 the meshes used to compute the solution for the low Péclet number (left panel) and the large Pécletnumber (right panel). In the last case, a local mesh refinement is used, as shown in Fig. 3 right panel, to suit well withthe boundary layer induced by the homogeneous Dirichlet condition. Fig. 4 presents the solution for the low Péclet number(left panel) and the large Péclet number (right panel) where the boundary layer thickness is about 1=100 of the unit length ofthe domain. To provide the convergence curves, we have considered successive finer meshes using the same local refinementprocedure close to the upper right corner. Table 1 and 2 show that we get an effective second-order, fourth-order and sixth-order L1-norm convergence with the P1, P3 and P5 reconstructions while the orders associated to the L1 norm are, in general,a little bit below. The numerical approximations do not suffer of any spurious oscillations or undesired overshoot even withlarge Péclet number thanks to the intrinsic upwind scheme used in the convective part. Another important point to underlineis that the P5 polynomial reconstruction manages to accurately capture the boundary layer even with a small number of cellswhile the P3 reconstruction requires more cells. The last line of Table 2 corresponds to the orders computed with the errorsof the first line and the fourth line involving two definitively different meshes. In the case of the low Péclet number case, a P5

reconstruction with 290 cells produces a better approximation than the P1 reconstruction with 16,858 cells and, by extrap-olation, a mesh of about 400,000 is required to obtain the same L1-error with a linear reconstruction.

We draw in Fig. 5 the convergence curves for the low Péclet number (top panel) and the large Péclet number (bottompanel) situations with the L1- and the L1-norm. The picture highlights the difficulty to provide an accurate approximationwith large Péclet number and the efficiency of the P5 reconstruction for this specific situation.

4.3. Deformed mesh

To check the robustness of the method and assess the capacity to handle complex meshes, we have carried out numericalsimulations with deformed meshes. An initial regular uniform structured mesh of size h ¼ 1=N is transformed into an irreg-ular one using the following mapping. For each inner vertex of coordinate ðx; yÞ, we move the point tox0 ¼ xþ ah cosðhÞ; y0 ¼ yþ ah sinðhÞ where h 2 ½0;2p� is an aleatory value (different for each node) and a 2 ½0;1=2� is thedeformation factor (also given in percentage). For example, a 15% deformation mapping means that one choosesa ¼ 0:15. Note that a deformation factor larger than 50% may produce overlapping cells hence we require that a < 1=2.

We have taken N ¼ 10;20;40;80 and experimented mapping with 15% and 45% deformation factors. We plot in Fig. 6 theconstant piecewise solution to visualize the two meshes. The low deformation mesh is rather close to the uniform situationwhereas the large deformation mapping induces an important form factor with large ratio between the cells. Similarly to theprevious example, numerical simulations have been performed with k ¼ 1; V ¼ ð5;5Þ and the normalizing constant is set toC ¼ 108. We present in Tables 3 and 4 the errors and convergence rates for the L1-norm and L1-norm while we display theconvergence curves in Fig. 7. We observe that the effective convergences are very close to the theoretical ones except for thesecond-order scheme which provides worse approximations. The noticeable points are that orders are preserved even withstrong deformed meshes and the errors slightly increase with the deformation factor. We report that the large deformationerrors are only 20% larger than the ones obtained with the small deformation mesh, which demonstrates the strong stabilityof the method.

Fig. 5. Convergence curves L1- and L1-norm for the low Péclet number (top) and the large Péclet number cases (bottom).

Fig. 6. Numerical approximation for low deformation (left) and large deformation (right).


4.4. Circular advection

To deal with the assessment of the scheme accuracy for a pure convective problem, we consider an inviscid fluid in thesquare X ¼ 0;1� ½2 which flows with a constant angular velocity setting V ¼ ð�y; xÞ and convects a smooth inflow passivequantity /Dðx;0Þ ¼ e�50ðx�0:5Þ2 from the bottom side. We prescribe the homogeneous Dirichlet condition on the right sideand outflow condition for the two other sides. The exact solution /ðx; yÞ ¼ e�50ðr�0:5Þ2 ; r2 ¼ x2 þ y2 is obtained as a quarter

Table 3L1 and L1 errors and convergence rates for low deformation case.



100 1.06e�02 . . . 9.45e�02 . . . 1.17e�03 . . . 7.53e�03 . . . 5.59e�04 . . . 2.64e�03 . . .

400 5.08e�03 1.1 4.14e�02 1.2 1.37e�04 3.1 1.95e�03 2.0 8.74e�06 6.0 1.40e�04 4.21600 1.55e�03 1.7 1.66e�02 1.3 1.02e�05 3.7 2.80e�04 2.8 2.42e�07 5.2 5.60e�06 4.66400 4.62e�04 1.8 4.85e�03 1.8 7.26e�07 3.8 2.64e�05 3.4 3.18e�09 6.2 1.18e�07 5.7

Table 4L1 and L1 errors and convergence rates for large deformation case.



100 1.46e�02 . . . 1.52e�01 . . . 1.79e�03 . . . 1.56e�02 . . . 5.88e�04 . . . 2.97e�03 . . .


Fig. 7. Convergence curves L1- and L1-norm for low deformation (top) and large deformation cases (bottom).


of revolution deriving from the boundary inflow condition. We use Delaunay meshes composed of triangles with a localrefinement around the circle r ¼ 0:5. The circular advection problem is a common test case but usually solved as the asymp-totic limit in time of a unstationary one involving time discretization whereas we here propose a specific strategy for sta-tionary problem.


We plot in Fig. 8 the mesh (left panel) and the numerical solution (right panel). In all the experiences we have performed,no over- or under-shoots have been observed even with the sixth-order scheme. We display in Fig. 9 the convergence curvesfor the L1- and L1-norms while Table 5 reports the errors and the convergence rates. Effective orders correspond to the ex-pected theoretical accuracy for the L1-norm but we do not achieve the optimal convergence rate for the L1 norm.

4.5. Pure diffusive problem with discontinuous coefficient

We now turn to the situation where the diffusion coefficient is discontinuous. Domains X1 ¼�0;1½��0;1=2½ andX2 ¼�0;1½��1=2;1½ share a common interface C ¼�0;1½�f1=2g and we assume that V1 ¼ V2 ¼ ð0;0Þ (pure diffusion problem)while k1 and k2 are constant positive values.

We consider a pseudo one-dimension problem setting the adiabatic condition kr/ � n ¼ 0 on the vertical boundary andwe prescribe the Dirichlet condition / ¼ T1 on y ¼ 0 and / ¼ T2 on y ¼ 1. Two conditions are experimented in the presentstudy.

� We first assume the linear heat transfer condition (problem Pf ) on C, namely

k1r/1 � nC ¼ k2r/2 � nC ¼ hð/2 � /1Þ: ð13Þ

One can check that functions� 2 � 2

/1ðx; yÞ ¼Ak1

1p

sinðpyÞ þ ak1

yþ T1; /2ðx; yÞ ¼Ak1

1p

sinðpyÞ þ ak2ðy� 1Þ þ T2

with

a ¼ h1þ h

2k1þ h

2k2

A1p

� 2 1k2� 1

k1

� þ T2 � T1

!
are the A-parameterized solution of the diffusion problem with source term f ¼ A sinðpyÞ.� We now substitute the linear heat transfer condition by the continuity of / across the interface C (problem Pc), that is,
k1r/1 � nC ¼ k2r/2 � nC; /2 � /1 ¼ 0: ð14Þ

In the same way, functions� 2 � 2

/1ðx; yÞ ¼Ak1

1p

sinðpyÞ þ ak1

yþ T1; /2ðx; yÞ ¼Ak1

1p

sinðpyÞ þ ak2ðy� 1Þ þ T2

with

a ¼ 11

2k1þ 1

2k2

A1p

� 2 1k2� 1

k1

� þ T2 � T1

!
are the solution of the diffusive problem with the source term f ¼ A sinðpyÞ.
All the simulations have been carried out with k1 ¼ 0:1; k2 ¼ 1, and h ¼ 1 when dealing with the heat transfer condition.In Fig. 10 we display on the left panel an elevation of the numerical approximation for the condition (13) and the right panel

Fig. 8. Circular advection: mesh and numerical simulation.

Fig. 9. Circular advection: convergence curves for the L1-norm (left panel) and the L1-norm (right panel).

Table 5L1 and L1 errors and convergence rates for the circular advection problem.



504 1.31e�02 . . . 6.33e�02 . . . 4.88e�03 . . . 2.31e�02 . . . 3.72e�03 . . . 1.29e�02 . . .

1876 2.53e�03 2.5 2.46e�02 1.4 1.89e�04 5.0 2.57e�03 3.3 3.47e�05 7.1 6.75e�04 4.57600 4.79e�04 2.4 5.14e�03 2.2 6.79e�06 4.8 8.84e�05 4.2 3.47e�07 6.6 1.44e�05 5.5

20,608 1.86e�04 1.9 2.87e�03 1.2 9.69e�07 3.9 1.80e�05 3.2 2.74e�08 5.1 1.23e�06 4.9

Fig. 10. Numerical approximation using the linear heat transfer condition (left) and the continuity condition (right).

Table 6L1 and L1 errors and convergence rates with the heat transfer condition



172 5.88e�02 . . . 2.20e�01 . . . 1.29e�03 . . . 6.32e�03 . . . 1.87e�05 . . . 1.26e�04 . . .



Table 7L1 and L1 errors and convergence rates with the continuity condition



218 1.28e+00 . . . 4.65e+00 . . . 3.84e�03 . . . 2.43e�02 . . . 4.09e�05 . . . 2.17e�04 . . .

544 3.94e�01 2.6 1.68e+00 2.2 2.26e�04 6.2 3.06e�03 4.5 1.44e�06 7.3 7.91e�06 7.21520 1.18e�01 2.3 4.56e�01 2.5 3.28e�05 3.8 4.09e�04 3.9 8.01e�08 5.6 6.22e�07 5.04116 5.17e�02 1.7 2.37e�01 1.3 3.61e�06 4.6 8.90e�05 3.0 3.51e�09 6.3 3.89e�08 5.6

Fig. 11. Convergence curves for the heat transfer condition (top) and the continuity condition (bottom).

Fig. 12. Numerical approximations using the linear heat transfer condition (left) and the continuity condition (right).



present the solution with the continuity assumption (14). Note that even with a discontinuous solution, we achieve the max-imal convergence order and no oscillations are observed in the vicinity of the discontinuity. The specific procedure we haveproposed gives very good results for both discontinuous coefficients and solutions when the discontinuity localization isknown.

Tables 6 and 7 provide the L1- and L1-norm errors with their respective convergence order for the heat transfer and thecontinuity condition respectively. Second-order, fourth-order, and sixth-order are almost achieved and the P5 reconstructionprovides an excellent convergence rate. An extrapolated assessment predicts that the second-order scheme should require39.000 cells to achieve the same error than the sixth-order scheme with just 218 cells. In the same way, the L1 error of2:5410�8 obtained with the sixth-order scheme on a 4116 cells mesh should be obtained with the second-order scheme witha mesh of about 13:109 of cells. Fig. 11 displays the convergence curves for the heat transfer condition (top panels) and thecontinuity condition (bottom panel) with the two norms L1 and L1.

Remark 4.3. Problem Pc is the limit case of problem Pf when h!1 but a simple calculation show that the error is of orderO 1

h

�. One can expect to accurately approximate solution of problem Pc using problem Pf with an h value large enough.

Assuming for example that the sixth-order scheme provides an error of order 10�6, one has to choose h � 106 which providesa linear system with a large conditioning number matrix. We conclude that it is not a good technique to compute anapproximation for problem Pc as a penalization of problem Pf when dealing with very high-order methods.

4.6. Convection–diffusion with discontinuous coefficients

The situation where both the velocity and the diffusion coefficient are discontinuous across interface C is more complex.The present study assumes continuity of the normal velocity for the sake of conservativeness. Notice that a discontinuousnormal velocity on edge eij � C results in a Dirac distribution as a source term and should be treated with a specific well-balanced method using non-conservative flux which is out of the scope of the present study.

We choose k1 ¼ 23; V2 ¼ ð0;0Þ and k2 ¼ 0:18; V1ðu;0Þ which guarantees a null flow across the interface. We propose anon-genuinely two-dimensional solution where the calculation is detailed for the sake of consistency. The solution will takethe following form:

Table 8L1 and

Nb o

262584

15204112

Table 9L1 and

Nb o

262584

15204112

/1ðx; yÞ ¼ A1eu

k1xyþ a1yþ T1; /2ðx; yÞ ¼ e

uk1

x½A2ðy� 1Þ þ B2ðy� 1Þ2� þ a2ðy� 1Þ þ T2:

Conservation of the flux at the interface yields

k1A1 ¼ k2A2 � k2B2; k1a1 ¼ k2a2;

while the linear heat transfer condition (2) across the interface provides

4k1A1 ¼ hðB2 � 2ðA1 þ A2ÞÞ; 2k1a1 ¼ hð2ðT2 � T1Þ � ða1 þ a2ÞÞ:

After some algebraic manipulations, one has

A2 ¼ �A1 2þ k1

k2þ 4

k1

h

� ; B2 ¼

k2A2 � k1A1

k2; a1 ¼

2ðT2 � T1Þ1þ k1

k2þ 2 k1

h

; a2 ¼k1

k2a1: ð15Þ

L1 errors and convergence order for convection diffusion problem with discontinuous coefficients: the transfer condition case

f cells P1 P3 P5


1.15e�01 . . . 8.48e�01 . . . 1.86e�03 . . . 1.24e�02 . . . 2.19e�05 . . . 1.56e�04 . . .

5.72e�02 1.9 5.25e�01 1.3 4.09e�04 4.1 2.38e�03 4.5 2.55e�06 5.8 2.44e�05 5.02.24e�02 2.0 2.04e�01 2.0 6.42e�05 3.9 5.30e�04 3.1 1.21e�07 6.4 1.24e�06 6.27.16e�03 2.3 8.27e�02 1.8 7.75e�06 4.2 9.10e�05 3.5 6.44e�09 5.9 8.01e�08 5.5

L1 errors and convergence order for convection diffusion problem with discontinuous coefficients: the continuity condition case.

f cells P1 P3 P5


1.87e�01 . . . 1.56e+00 . . . 1.68e�03 . . . 1.07e�02 . . . 2.07e�05 . . . 1.32e�04 . . .

9.15e�02 1.9 7.50e�01 2.0 3.81e�04 4.0 2.01e�03 4.5 2.70e�06 5.5 1.96e�05 5.23.11e�02 2.2 2.72e�01 2.1 5.49e�05 4.0 4.50e�04 3.1 1.38e�07 6.2 1.01e�06 6.21.18e�02 1.9 1.30e�01 1.5 6.55e�06 4.2 7.70e�05 3.6 5.59e�09 6.4 6.76e�08 5.4

Fig. 13. Convergence curves L1-norm (top), L1-norm (bottom).

Fig. 14. Cooling stage case study geometry (dimensions in mm).

Fig. 15. Cooling stage case study mesh (dimension in m).


Fig. 16. Temperature distribution in the cooler (�C).


Note that the expressions depend on the free parameter A1 that we can fix arbitrarily. The source term is given by f1 ¼ 0 inX1 and f2 ¼ �k2e

uk1

x 2B2 þ uk1

� 2ðy� 1ÞðA2 þ B2ðy� 1ÞÞ

� �in X2. For the sake of simplicity, we use the exact solution as a Dirich-

let condition on the boundary.We revisit the previous solution prescribing a continuous transition. Conservation of the flux at the interface yields

k1A1 ¼ k2A2 � k2B2; k1a1 ¼ k2a2;

while the continuity across the interface provides the relations

2A1 þ 2A2 ¼ B2; a1 þ a2 ¼ 2ðT2 � T1Þ:

After some algebraic manipulations, one has

A2 ¼ �A1 2þ k1

k2

� ; B2 ¼ 2A1 þ 2A2; a1 ¼

2k2

k1 þ k2ðT2 � T1Þ; a1 ¼

2k1

k1 þ k2ðT2 � T1Þ: ð16Þ� �

Notice that f1 ¼ 0 in X1 and f2 ¼ �k2eu

k1x 2B2 þ u

k1

� 2ðy� 1ÞðA2 þ B2ðy� 1ÞÞ in X2 as before. Moreover, one can check that

we formally recover the coefficients (16) for the continuous transition taking h!1 in (15).We plot in Fig. 12 the numerical approximations with the transfer condition (left panel) and the continuity condition

(right panel). We emphasize that the solutions do not suffer of any spurious oscillations in the vicinity of the interface.We report in Tables 8 and 9 the error in L1- and L1-norm for the heat transfer condition and continuous transition respec-tively. In both cases we obtain the expected convergence order and the P5 reconstruction provides an effective sixth-orderscheme. We display in Fig. 13 the convergence curves both for the linear heat transfer condition (top) and the continuitycondition (bottom).

4.7. Application to the cooling stage of polymer extrusion and preconditioner performance

To conclude the section, we consider the concrete application of the cooling stage of polymer extrusion (see [32–34] forthe polymer flow problem and the cooler set-up design) and assess the preconditioning matrix performances. As illustratedin Fig. 14, the polymeric slab, extruded at high temperature, is cooled in contact with a metallic part (the calibrator), whichhas temperature controlled by a fluid that flows inside the cooling channels. A polymer flows in domain X2 in contact with ametallic cooler X1 where temperature continuity is prescribed at the contact interface (see Fig. 14). The relevant governingequations for this problem are

�r � ðkmr/Þ ¼ 0 in X1; r � ðqCpV/� kpr/Þ ¼ 0 in X2;

with q ¼ 1200 kg/m3 the polymer density, Cp ¼ 1000 J/(kg K) the polymer specific heat capacity, V ¼ ð0:01;0Þm/s the poly-mer velocity, kp ¼ 0:18 W/(m K) the polymer thermal conductivity, and km ¼ 23 W/(m K) the metal thermal conductivity.The cooling process is ensured by prescribing the temperature (Tw ¼ 10: C) at the cooling channel’s surface. On the otherhand, we fix the inflow polymer temperature Tp ¼ 200: C, while adiabatic condition are prescribed for the remainingboundaries.

As a first test, a mesh with 17,967 cells is used to perform the computation (see Fig. 15) and we use a P5 polynomialreconstruction. Fig. 16 represents the temperature distribution obtained while we display in Fig. 17 the heat flux normj krTj. These results evidence the role of the cooling channels, whose aim is to remove heat from the system. No spu-rious oscillations are detected in the vicinity of the interface and the temperature map is similar to the one obtained in[33].

To check the preconditioning matrices performance, we have carried out simulations with a 23,616 triangles mesh usingthe second-, fourth- and sixth-order schemes, with the two candidate preconditioning matrices, where iterations were per-

Fig. 17. Heat flux distribution in the cooler (W/m2).

Fig. 18. Residual histogram curves with three preconditioning matrices for the second-order (top), fourth-order (middle) and sixth-order scheme. Thepreconditioning matrix are: the identity matrix (red), diagonal Patankar-like matrix (green), and the full Patankar-like matrix (blue). (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article.)


formed till the initial residual was reduced to 6 orders of magnitude. Experiments with the identity matrix (no precondition-ing) was also done for comparison purposes. For this problem, a restart procedure is activated when the difference betweentwo successive residuals is lower to 2� 10�3 of the current residual.


We display in Fig. 18 the residual histogram for the three preconditioning matrices tested with the second-, fourth-, andsixth-order schemes. Clearly the preconditioning matrix is fundamental to dramatically reduce the computational effort andthe AyP provides the most effective residual reduction (a lower number of iterations to reduce the initial residual to 6 orders ofmagnitudes). The low convergence rate obtained without preconditioning, when compared with the remaining, is an addi-tional evidence of the preconditioning relevance.

5. Conclusions

We have presented a new very high-order finite volume method for steady-state linear convection diffusion problems.The scheme achieves an effective sixth-order accuracy for a large panel of situations: low and large Péclet numbers, diffusionand convection with discontinuous coefficients or discontinuous solutions. We show that the method is robust, provides anM-matrix for pure diffusion problems and does not generate spurious oscillations in the vicinity of the discontinuity, thanksto the multidomain strategy adopted. We have restricted the study to the linear case, to assess the ability of the new recon-structions to achieve very high-order approximations, especially the polynomial functions associated to the boundary, whichprovide the diffusive contributions and contain the Dirichlet condition.

Two main extensions are now under consideration. An accurate treatment of the boundary, when dealing with smoothboundaries, will be designed to preserve the sixth-order rate. Such a topic is not well-developed in the finite volume contextand the diffusion case (see [18] for instance). On the other hand, nonlinear situations will be investigated, where a limitingstrategy will handle discontinuity situations to avoid unphysical oscillations (small diffusion with large gradient of the solu-tion for instance). The MOOD strategy [24–26] is a good candidate to control the local discontinuities while preserving theaccuracy in the smooth regions.

Acknowledgments

The authors research was financed by FEDER Funds through Programa Operacional Factores de Competitividade – COM-PETE and by Portuguese Funds through Fundação para a Ciência e a Tecnologia, within the project PTDC/MAT/121185/2010.

The first, second, and the fourth author research was financed by FEDER Funds through Programa Operacional Factores deCompetitividade – COMPETE and by Portuguese Funds through Fundação para a Ciência e a Tecnologia, within the ProjectPEst-C/MAT/UI0013/2011 and the third author within the Project PEst-C/CTM/LA0025/2013.

The first, second, and fourth authors research was financed by the Programa Pessoa Cooperação Transnacional Portugal/França Fundação para a Ciência e a Tecnologia under the reference 130631467126259.

The authors would like to acknowledge N.D. Gonçalves for providing the geometry files used for the cooling stage casestudy.

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A sixth-order finite volume method for multidomain convection–diffusion problem with discontinuous coefficients