A review of Residual distribution schemes for hyperbolic and parabolic problems : the july 2010 state of the art

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and parabolic problems : the july 2010 state of the art

Rémi Abgrall

To cite this version:

Rémi Abgrall. A review of Residual distribution schemes for hyperbolic and parabolic problems : the july 2010 state of the art. [Research Report] Inria. 2011, pp.39. �inria-00526162v2�

perbolic and parabolic problems : the July 2010 state of the art.

R. Abgrall

Team Bacchus, Institut de mathmatiques de Bordeaux INRIA and University of Bordeaux

33 405 Talence cedex

Abstract. We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special em- phasis is put on the structure of the computational stencil. We provide their con- nections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these meth- ods to parabolic problems. We also draw some research perspectives.

Key words: Hyperbolic problems, Navier Stokes equations, Unstructured meshes, Residual dis- tribution methods

1 Introduction.

The numerical simulation of compressible flow problems, or more generally speaking, of partial differential equations (PDEs) of hyperbolic nature, has been the topic of a huge literature since the seminal work of von Neuman in the 40’s. Among the “hot” topics of the field has been, since the works of Lax, Wendroff, Godunov, Mc Cormack, van Leer, Roe, Harten, Yee and Osher, to give a few names, the development of robust, parameter free and accurate schemes. Among the most successful methods one may quote the van Leer’s MUSCL method [42] and modified flux approach of Roe. These techniques are only second order accurate. The accuracy can be improved via the ENO/WENO methods by Harten, Shu and others.

The emergence of modern parallels computers, another concern has emerged: what about accuracy and efficiency ? Indeed, it is now important to develop robust algorithms that scale correctly on parallel architecture. This can be achieved more or less easily if the stencil of the numerical scheme is as compact as possible. Good candidates are the schemes relying on finite element technology, such as the Discontinuous Galerkin (DG)

http://www.global-sci.com/ Global Science Preprint

methods [14] or the stabilized continuous finite element(CFE) methods [22, 23]. In these methods, the numerical stencil is the most possible compact one.

In these notes, we discuss in some details of another class of numerical schemes, the so-called Residual Distribution schemes (RD for short), also denoted by Fluctuation Split- ting schemes. The history of these schemes can be traced back to the work of P.L. Roe [36]

and even his famous 1981 paper [37] where he does not define a finite volume scheme but a true residual distribution scheme. Indeed, the first RD scheme ever was probably presented by Ni [26]. The idea was to construct a scheme with the most compact compu- tational stencil that can ensure second order accuracy. This scheme had some similarities with the Lax Wendroff one.

If these RD share many similarities with more established schemes such as the SUPG scheme by Hughes and coworkers [20–22], the driving idea is (i) to introduce the up- winding concept, (ii) to manage such that a provable or a practical maximum principle is achieved without any parameter to tune. In our opinion, (ii) is the most important feature.

In Roe’s paper and the first RD papers, the main idea was to introduce upwinding into the numerical formulation of the problems, coupled in a very clever way, with a technique to reach second order accuracy for steady problem. This has been presented in a series of papers and VKI reports, see e.g. [15–17,29,40,41]. Two schemes had emerged at the time : the N scheme by Roe and the PSI scheme by R. Struijs, see [17]. The first one is probably the optimal first order strategy for scalar problems using triangular meshes, the second one the best second order scheme on these type of meshes, for steady problems again. When dealing with systems or non triangular meshes, the situation became more complex, and it appeared that the upwinding concept had to be relaxed a bit.

Since the early days, many contributions have been given. Among the issues, two are more difficult because they do not cast a priori naturally in the original RD framework.

Let us mention the approximation of unsteady problems where, in addition of the work of the authors and his collaborators, as Mario Ricchiuto (INRIA), we have to mention the work of Degrez et al [25], De Palma et al. [30, 39]. An interesting contribution on the approximation of viscous problem is the work of H. Nishikawa [27, 28]. Last, and up to our knowledge the first contribution on higher than second order accurate RD scheme is due to Caraeni [12, 13], as well as early work on unsteady and viscous problem. One of the main differences with the approach emphasized in this paper is that Caraeni’s schemes are not as compact as here

This paper presents the author’s personal view of what is the current status of RD scheme for steady problems. In a first part, we present a reinterpretation of standard finite volume schemes and show on a simple example how maximal accuracy can be reached with a minimal stencil. In the second part, we present a general framework to describe what a RD scheme is and provide some examples. Then we give a very formal variational formulation and some connection with more established schemes such as the Discontinuous Galerkin schemes or the stabilized continuous finite element methods. In a second part, we discuss a systematic way of getting a non oscillatory scheme, without

tunable parameter, even in the system case. Numerical examples are given for illustra- tion. The last section is devoted to some extensions, in particular the unsteady case and the viscous case.

2 Some remarks about finite volume schemes

2.1 Two formulations of finite volume schemes.

Let us start with a simple example. We consider the following problem

∂u

∂t +^∂f(u)

∂x =0 (2.1)

with initial and boundary conditions that we do not specify for the moment. Using a regular mesh (xj=j∆x), this problem is discretised by a simple finite volume scheme

∆xuⁿ_i⁺¹−uⁿ_i

∆t +Fi+1/2−Fi−1/2=0 (2.2) whereFj+1/2 is the numerical flux at the cell interfacexj+1/2=^xj+₂^xj+1. It depends on the local cell averages of the solution{ul}_l^j+₌^p_j−p, where

uj≈ Rx_j+1/2

x_j−1/2u(x,t)dx

∆x . We can rewrite (2.2) as

∆xuⁿ_i⁺¹−uⁿ_i

∆t +φ_i⁻_+1/2+φ_i⁺_−1/2=0 (2.3) where we have set

φ_i⁻_+1/2=F_i+1/2−f(ui), φ⁺_i−1/2=f(ui)−F_i−1/2. In each interval[xi,xi+1], we have introduced the “residuals”

φ_i⁻_+1/2=Fi+1/2−f(ui), φ⁺_i+1/2=f(ui+1)−Fi+1/2. (2.4) The two formulations (2.2) and (2.3) are of course equivalent.

If the numerical scheme F_j+1/2 is consistent with the continuous one, and depends continuously of its arguments, assuming in addition some stability assumptions, the Lax Wendroff theorem states that the solution of (2.2) converges to a weak solution of (2.1). In the proof of this theorem, the key algebraic argument is thatF_i+1/2−F_i−1/2is a difference of flux. Considering now (2.4), this argument is translated into the relation

φ⁻_i+1/2+φ⁺_i+1/2=f(ui+1)−f(ui) =

Z x_i+1

xi

∂f(u^∆x)

∂x dx (2.5)

whereu^∆x=uixi+1−x

∆x +ui+1xi−x

∆x . On can show, see for example [6] for a more complex case, that under the assumptions of the Lax Wendroff theorem (stability assumptions, and continuous dependency of the residuals with respect to their arguments), that the solution of (2.3) converges to a weak solution of (2.1).

2.2 About accuracy.

The goal is to construct schemes of the type (2.3)-(2.5) that have the most possible compact stencil with the maximum accuracy. We show here that there is some hope. For example, second order accuracy can be obtained with a 3 point stencil (instead of 5 for a standard high order scheme). This is done in two steps. We first consider the steady version of (2.1) and then extend the method to the unsteady case. Of course the steady version of (2.1) is trivial, but is rather enlightening to consider the following problem

u^′=λu x∈]0,1]

u(0) =1. (2.6)

The solution of (2.6) isu(x) =e^λx.

If one wishes to approximate (2.6) by an upwind finite volume, a natural formulation is

Fi+1/2−Fi−1/2=λ Z x_i+1/2

xi−1/2

u(x)dx≈^∆xλui.

Note that the source term is approximated with second order accuracy, andFi+1/2=ui, Fi−1/2=ui−1. The scheme is

ui−ui−1=^∆xλui, u0=1. (2.7) We obtainui= 1−λ^x_iⁱi

: ifiis chosen so thatxi=i∆x isfixed,ui−u(xi) =¹₂e^λxixiλ²∆x+ O(^∆x²): the convergence is only first order.

Consider now the scheme

ui−ui−1−^λ∆x

2 (ui+ui−1) =0 withu0=1. (2.8) We get

ui=

1+^λ∆x₂ 1−^λ∆₂^x

i

so that wheniis chosen withxi=i∆x fixed, u_i−u(x_i) = ¹

12e^λxiλ³x_i∆x²+O(^∆x⁴)

which shows so the convergence is second order. The scheme (2.8) can be interpreted in the residual distribution framework. To do that, we define the total residual by

φi+1/2=

Z x_i+1

xi

(u^∆x)^′−λu^∆x)dx=ui+1−ui−λui+ui+1

2

and the sub-residuals by

φ_i⁻_+1/2=φi+1/2, φ_i⁺_+1/2=0 i.e., we distribute on the downwind vertex of the cell[xi,xi+1].

This simple example shows that one can maximize accuracy with the smallest stencil.

This is precisely the philosophy that is pursued by the Residual Distribution schemes, with the goal of deriving non oscillatory schemes. We now describe what these schemes and then give connections with DG and CEM schemes.

3 Residual distribution schemes.

This section is devoted to a short description of RD schemes, and we specialize to scalar problems though this point, at this level is insignificant.

3.1 The model problem.

We first consider the steady problem

div f(u) =0 inΩ⊂^Rd (3.1a)

subjected to Dirichlet boundary conditions on the inflow partΓ⁻ofΓ=∂Ω,

u=ginΓ⁻ (3.1b)

IfM∈^Γand~_nis the outward unit vector atMofΓ, the inflow boundary is defined as Γ⁻={M∈^Γ,∇_uf(u(M))·~_n<₀}

3.2 Approximation space.

The domainΩis triangulated by a conformal mesh, the triangulation is denoted byT_h. The elements of this triangulation are triangles and quads in 2D, or tetrahedrons in 3D.

Other types of elements could certainly be tackled, but this has not yet been done. The elements ofT_h are denoted by {Ki}_i=1,ne and the vertices are denoted by{Mi}_i=1,ns. In most cases, we deal with one generic elementK; since there is no ambiguity, the vertices are denoted byi=1,nKwherenkis the number of vertices inK. The approximate solution of (3.1) will be sought for in the space

V^h={ucontinuous inΩ_h, for anyu|Kis polynomial of degree r.}.

Inddimensions, a polynomial of degreeris defined byn^d_r=C_d^d_+rcoefficients, i.e. ^(r+1)(₂^r+2) in 2D and ^(r+1)(r+₆^2)(r+3)in 3D. This means that a polynomial is uniquely defined if an uni- solvant set of points of cardinaln^d_ris given. In the case of triangles/tetrahedrons, the stan- dard Lagrange points , for example by their barycentric coordinates(_rⁱ,_r^j,^k_r)_{i,j,r≥0,i+j+r=d}

T_h

~_λ Γ⁻

Ω

Figure 1: A typical mesh

in the case of a triangle and(_rⁱ,_r^j)_i,j≥0,i+j≤rfor a quad when it is mapped onto[0,1]². The Lagrange points are the degrees of freedom at which an approximation ofuis sought for.

In all cases, the elements ofV^hcan be written as a linear combinations of basis functions, v=

∑

σ

vσϕσ,

where{σ}is the set of degrees of freedom and the basis functions satisfyϕσ(σ^′)=δ^σ_σ^′. We assume that the basis functions have a compact support, this is always true in practice.

In the case of the Lagrange interpolation, the basis functions are the standard Lagrange functions.

The class of triangulations that we consider are regular in the finite element meaning, i.e. there is a constantCT such that ifρKis the ratio of the outer diameter ofKto the inner diameter ofK(soρK≥1),

maxK∈T_hρK≤CT. (3.2)

As classical, the parameter “h” inT_h refers to the maximum of the diameters of the ele- ments contained inT_h.

3.3 Numerical discretisation.

The approximation is done in two steps. For any elementK, we define the total residual φK(u^h):=

Z

∂Kf(u^h)·~_{ndσ (3.3a)}

which is splitted into sub-residualsφ^K_j , one for each degree of freedomσj inK. Note that f(u^h)in (3.3a) can be replaced by the Lagrange interpolation of the flux f, this leads to quadrature free versions of RD schemes. Since there is no ambiguity, we denote these de- grees of freedom either byσj,j=1,...,nKor byj,j=1,...,nKor simply byσ,σ∈Kdepending on the context. The sub-residuals are contrainted by theconservationrelation

∑

σ∈K

φ_σ^K(u^h) =φ^K(u^h) (3.3b)

The scheme writes : findu^h∈V^hsuch that for any degree of freedomσ6∈^Γ−,

∑

K,σ∈K

φ_σ^K(u^h) =0 (3.3c)

while on the boundaryΓ⁻we set

u^h(σ) =g(σ). (3.3d)

Of course the problem (3.3) is in general a (very) non linear problem. Hence it is in practice solved by an iterative technique. We later rapidly come back to this point.

There are a couple of general results which explain the type of structure the residuals and sub-residuals should have in order to guaranty accuracy and convergence to a weak solution of (3.1), if the method converges.

3.4 Some examples.

In order to explain the relation (3.3), we provide three examples: the Galerkin scheme, the N scheme and the SUPG scheme. In each case, we consider f(u) =~_λuto simplify.

3.4.1 The Galerkin scheme.

Though unstable, this case is interesting. We consider a test functionϕ^h∈V^hthat vanishes onΓ⁻. The Galerkin formulation writes (withΓ⁺:=∂Ω−^Γ−)

0=

Z

Ωϕ^hdiv f(u)dx=

Z

∂Ωϕ^hu^h~_λ·~_ndl−

Z

Ωu^h~_λ·∇ϕ^hdx

=

∑

K

Z

∂Kϕ^hu^h~_λ·~_ndl−

Z

K∇u^h~_λ·ϕ^hdx

!

It is sufficient to specialize this relation to the basis functionsϕσ. Denoting φ^K,c_σ =

Z

∂Kϕ^h_σu^h~_λ·~_ndl−

Z

K∇u^h~_λ·ϕ^h_σdx (3.4) Using these notations, we see that (3.3b) holds true as well as (3.3c) and (3.3d): the Galerkin scheme can be seen as a RD scheme.

3.4.2 The SUPG scheme

The Galerkin scheme is notoriously unstable, one of its stabilized version is the SUPG scheme of Hughes and co-workers (see for example [22]). Instead of considering (3.4),

we consider (remember f(u) =~_λuhere) 0=

Z

Ωϕ^hdiv f(u)dx+

∑

K

hK

Z

K

~_λ·∇ϕ^h

τ

~_λ·∇u^h

dx

=

∑

K

Z

∂Kϕ^hu^h~_λ·~_ndl−

Z

K∇u^h~_λ·ϕ^hdx +hK

Z

K

~_λ·∇ϕ^h

τ

~_λ·∇u^h

dx

!

(3.5)

Specializing to the basis function, we introduce the residuals φ^K_σ=

Z

∂Kϕ^h_σu^h~_λ·~_ndl−

Z

K∇u^h~_λ·ϕ^h_σdx +hK

Z

K

~_λ·∇ϕ^h_σ

τ

~_λ·∇u^h

dx, and again (3.3b) holds true as well as (3.3c) and (3.3d).

3.4.3 The N scheme

The N (for Narrow) scheme is due to P.L. Roe [36]. Here, all the elementsKare triangles in 2D and tetrahedrons in 3D. The goal is to define an upwind RD scheme that generalizes the classical upwind scheme in 1D. The first remark is that, since again f(u) =~_λu,

φK(u^h) =

Z

K

~_λ·∇u^hdx=

∑

σ∈K

k^K_σuσ, k^K_σ=

Z

K

~_λ·∇ϕσ=¹ 2

~_λ·~_n^K_σ

where~_n^K

σ is the scaled inward normal to the opposite side of σ in K. We note that the parametersk^K_σ sum up to 0, and a careful examination indicates that their sign is a good indication of whether or not the vertexσ is upwind or downwind with respect to the elementK. Hence, we look for a Residual scheme of the type

φ_σ^K:=max(k^K_σ,0)

uσ−u_e

(3.6a) whereueis evaluated so that (3.3b) holds. We easily get

e u=

∑

σ^′∈Kmin(k^K_σ^′,0)uσ^′

∑

σ^′∈K

min(k^K_σ′,0) ^{. (3.6b)}

Note that if~_λ6=0, ∑

σ^′∈Kmin(k^K_σ′,0)6=0 because ∑

σ^′∈Kk^K_σ′=0. It can be shown that the solutions of (3.3c)-(3.3d) satisfy a maximum principle, see for example [38].

4 Design properties of RD schemes

Most of the results here are described for scalar problems, but extends naturally to sys- tems. Special comments are given for systems in order to explain what is done to fulfill the accuracy constraints in practice.

4.1 Convergence to a weak solution.

We have the following result that has been shown in [6].

Proposition 4.1. We consider a family of triangulations that satisfy (3.2) and such that h→0. Assume that the sub-residuals depend continuously on u^h, that there exists a constantCindependent ofhsuch that

maxσ∈T_h|u^h(σ)| ≤C

and a functionv∈L²(Ω)such that a sub sequenceu^hnk→vinL²(Ω)whenk→+∞. Then vis a weak solution of (3.1)

The key argument is the conservation relation (3.3b). In (3.3a), the integral is generally obtained by numerical quadrature and the result is independent of the numerical quadra- ture, provided that on any edge/face ofT_h, the set of quadrature points only depend on the edge/face and not on the particular element this edge/face is part of.

4.2 Accuracy constraints.

On an unstructured mesh, it is difficult, if not hopeless, to derive an error analysis via Taylor expansion, because the mesh has in general no geometrical symmetries. Hence, it is better to rely on a weak form of the truncation error. Consider ϕa C¹ function onΩ with||ϕ||∞≤1. This inequality is set up for scaling purpose only. We define the truncation error forw^h, the interpolate of the exact solution of (3.1) (assuming it is smooth enough) by

E_h(u) = max

ϕ∈C¹(Ω),||ϕ||∞≤1,||∇ϕ||∞≤1

∑

σ∈T_h

ϕ(σ)

∑

K,σ∈K

φ_σ^K

!

(4.1a) and the scheme is p-th order accurate if there exists a constantCindependent ofhsuch that

E_h(u)≤Ch^p. (4.1b)

There is a simple construction that permits, formally at least, to fulfill (4.1b). It relies on the use of the structure of (3.1): it is a steady problem. The case of time dependent problem will be considered later. The key remark is that if for any σ and K, the sub- residuals (evaluated for an interpolationw^hof orderk+1 of the exact solution, assuming it is smooth enough) satisfy

|φ_σ^K(w^h)| ≤Ch^k+d (4.2)

whereCis independent ofT_h satisfying (3.2), then (4.1b) holds for p=k+1. Again the proof is given in [6], and we recall it shortly. We introduce the Galerkin residuals (3.4),

φ^T,c_σ =

Z

Kϕ^h_σdiv f(u^h)dx and we have

∑

σ∈T_h

ϕ(σ)

∑

K,σ∈K

φ^K_σ

=

∑

K

∑

σ∈K

ϕ(σ)φ_σ^T

=

Z

Ωϕ^hdiv f(u^h)dx+

∑

K

∑

σ∈K

ϕ(σ)(φ^K_σ−φ^K,c_σ )

The next step is to see that ∑

σ∈Kφσ^T,c=φ^Kso that we have

∑

σ∈K

ϕ(σ)(φ_σ^K−φ_σ^K,c) = ¹ nK!

∑

σ,σ^′∈K

(φ(σ^′)−φ(σ))(φ_σ^K−φ_σ^K,c)

Then, we make the following remark: if the exact solution ofsteadyversion (3.1) is smooth enough, then for anyσandK

φ^K,c_σ (w^h) =O(h^k+d)andφ^K(w^h) =O(h^k+d).

Let us look at the first relation. The second one is the sum overσ∈Kof these relations.

We have

φ^K,c_σ (w^h) =

Z

Kϕ^h_σdiv f(w^h)dx=

Z

Kϕ^h_σdiv

f(w^h)−f(u)

dx

=

Z

∂Kϕ^h_σ

f(w^h)−f(u)

·~_ndx−

Z

K∇ϕ^h_σ

f(w^h)−f(u)

dx

=O(h^d−1)×O(h^k+1)+O(h^d)×O(¹

h)×O(h^k+1)

=O(h^k+d)

To get the second line, we explicitly use the fact thatR

Kϕ^h_σdiv f(u)dx=0 because the problem is steady, the second line comes from the Gauss theorem, the third line uses that fact thatf is Lipschitz continuous,w^h−u=O(h^k+1)and the regularity assumption of the mesh.

Thanks to this,

E^′=

∑

K

∑

σ∈K

ϕ(σ)(φ^K_σ−φ_σ^K,c)

=O(h^k+1).

The proof is very simple. We have E^′=

∑

K

∑

σ∈K

ϕ(σ)(φ^K_σ−φ^K,c_σ )

=

∑

k

1 nK!

∑

σ,σ^′∈K

(ϕ(σ)−ϕ(σ^′))(φ^K_σ−φ_σ^K,c)

=NK×nK×O(∇ϕ)×h× O(φ_σ^K)+O(φ^K,c_σ ) because

1. |ϕ(σ)−ϕ(σ^′)| ≤ChK≤Chsince ϕisC¹:Conly depends on||∇ϕ||∞,

2. The mesh is regular so that the numberNKof elements isO(h^−d),nK is fixed, 3. Last,φσ^K,c=O(h^k+d)andφ_σ^K=O(h^k+d), hence

E^′=O(h^−d)×O(h)×O(h^k+d) =O(h^k+1). The last remark is that

Z

Ωϕ^hdiv f(w^h)dx=−

Z

Ω∇ϕ^h(f(w^h)−f(w))dx=O(h^k+1)

because||∇ϕ^h|| ≤Cuniformly. Here we have neglected the boundary conditions to sim- plify (abusely) the situation. The calculation can be made rigorous by a proper and sim- ple approximation of the boundary conditions, see [31] for details. This shows that if φ^K_σ=O(h^k+d)then

E_h(u)≤Ch^k+1 and the scheme is (formally)k+1-th order accurate.

This analysis leads to residuals of the form

φ^K_σ=β^K_σφK (4.3)

where the family {β^K_σ}_σ,K is uniformly bounded whenh→0. In the rest of the text, we denote the schemes of the type (4.3) as “unfiltered” RD schemes for reasons that are explained later in this paragraph.

In the next paragraph, we discuss the construction of scheme of the form (4.3) that are both formally high order accurateand L^∞ stable. In many cases, one can see experimen- tally that the schemes (4.3) are over-compressive. This can be cured if one adds dissipa- tion. This can be done without violating (4.2) by adding a selected form of dissipation, namely

φ_σ^K=β^K_σφK+θhK

Z

K ∇uf(u^h)·∇ψσ

τ ∇uf(u^h)·∇u^h

dx (4.4)

whereθis a positive parameter. This form of dissipation is reminiscent of the stabilization term of the SUPG scheme [22] but here, as shown later, it plays the role of filter. In practice, the positive parameterτis set to

τ=

∑

i: vertices of K

max(∇uf(u^h)∇^Ψh_i,0) −1

(4.5)

and the integral in (4.4) is replaced by a quadrature-like form. In [5], we study in detail what are the minimum requirements that this quadrature-like formula should met in term of weights and quadrature points. The design criterion is that the term (4.4), or its discrete counterpart, should be strictly dissipative for any functionv^hsuch that∇uf(u^h)·

∇v^h6=0 on enough points inK. It turns that very few points are needed to fulfill these requirements, for example only 3 in 2D for quadratic approximations instead of at least 6 if one wishes to integrate (4.4) exactly. In 3D, the CPU saving is even more important.

Details can be found in [5]. In (4.5), we only consider the vertices ofK, and theΨ_is are the lowest order finite element constructed onK : linear polynomials for triangles and tets, Q¹ for quads and hex, etc. Lastu^h is the arithmetic average on the degrees of freedom in K. The schemes where (4.4) or (4.5) are added to (4.3) are denoted as “filtered” RD schemes.

Remark 4.1(About the effective accuracy). In practice, we are never able to exactly satisfy (3.3c) for any degree of freedom, but

∑

K,σ∈K

φ_σ^K(u^h) =εσ (4.6)

The previous truncation error analysis leads to E_h(u)≤Ch^k+1+

∑

K

εσ

ans the same analysis shows that we need to have

maxσ |εσ|=O(h^k+d+1). (4.7) This is why it is of extreme importance that the convergence of the iterative scheme used to evaluate the discrete solutionu^hcan be pushed far enough. This is precisely the role of the term (4.4) or its discrete counterpart. If this term is not present, the scheme will not converge, see [9].

4.3 Getting both accuracy and stability.

All the known RD schemes have the form φ^K_σ=

∑

σ^′∈K

cσσ^′(uσ−u^′_σ).

It is well known that ifcσσ^′≥0, and if a solution of (3.3) exists, it satisfies a maximum principle. Hence, we are going to construct schemes of the form (4.3) with positivecσσ^′. This is done in two steps.

• First step. We construct a family of sub-residuals that ensures first order accuracy and stability inL^∞. The simplest choice is an extension of the local Lax Friedrichs scheme:

φ^K,LxF_σ =^φ

K

nK

+α uσ−uK

(4.8)

with

u=^∑σ∈Kuσ

nK andα≥max

K ||∇_uf(u^h)||.

These choices guaranty that the scheme isL^∞stable, and more precisely we have cσσ^′= ¹

nK

_Z

T∇_uf(u^h)·∇ψσ^′dx+α

• We define(φ^K_σ)^⋆=β^K_σφ^Kwith

β^K_σ= ^max(0,^φσK,LxF_φK )

∑

σ^′∈K

max(0,^φ

K,LxF σ′

φ_K )

. (4.9)

This is one of the many choices that guaranties (φ^K_σ)^⋆=∑_σ′∈Kec_σσ^′(uσ−u_σ^′) with e

cσσ^′≥0. It is constructed following:

(φ_σ^K)^⋆= (φ_σ^K)^⋆ φ^K,LxFσ

φ_σ^K,LxF

∑

σ^′∈K

(φ^K_σ)^⋆ φ^K,LxFσ

c^LxF_σσ′(uσ−u_σ^′)

Then we setec_σσ^′=(φ^K_σ)^⋆

φ_σ^K,LxFc^LxF_σσ′ which is positive if (and only if)(φ^K_σ)^⋆×φ^K,LxFσ ≥0 that is,

β^K_σφσ^K,LxF

φK

≥0.

The relation (4.9) is obtained by satisfying these relations for anyσ∈K.

4.4 Extension to systems.

In the case of systems,

divf(u^h) =0

the generalization is straightforward: no modification is needed except the way the co- efficientsβ^K_σ is evaluated. We first note that sinceφKandφ_σ^LxFare vectors, the construc- tion (4.9) is meaningless. This is why we rely on a characteristic decomposition of the

total and sub-residuals. More precisely, we consider a direction ~d, the left and right eigen-vectors of K^~_d:=∇f(u)·~d. They are denoted, respectively, by {r_ξ}_ξeigenvalues of A

and{ℓ_ξ}_ξeigenvalues ofK_d~. By construction, we haveℓ_ξ(r_ξ^′) =δ_ξ^ξ′. Consider a set of first order residuals{φ^K,Lσ }_σ∈Kwith

∑

σ∈K

φ_σ^K,L=φK=

Z

∂Kf(u^h)·~_ndl.

An example is given by the Lax Friedrichs residuals, φ^K,LxF_σ = ¹

nK

φK+αK(uσ−u) withu= ¹ nK

∑

σ∈K

uσ. We decompose the residualsφ^K,Lσ onto the eigen-basis,

φ^K,L_σ =

∑

ξeigenvalues ofK_d~

ℓ_{ξ φ}^K,L_{σ rξ (4.10a)}

By construction, we have, for anyξ,

∑

σ∈K

ℓ_{ξ φ}^K,L_σ =ℓ_{ξ φK (4.10b)}

and we remark that the characteristic ℓ_{ξ φ}^K,L_σ are scalar quantities. We can apply the same technique as in the scalar case to them. For example, using (4.9), we define

ℓ_{ξ φ}^K,L_σ ^⋆=β^K,ξσ ℓ_{ξ φK (4.10c)} and then the high order residuals are

φ_σ^K,⋆=

∑

ξeigenvalues ofK_d~

ℓ_{ξ φ}^K,L_σ ^⋆_{rξ. (4.10d)} The last step, as in the scalar case, is to add a dissipation term, as in (4.4). By analogy, the final residual is

φ_σ^K=φ^K,_σ^⋆+θhK

Z

K

∇uf(u^h)·∇ϕσ

τ

∇uf(u^h)·∇u^h

dx (4.10e)

The matrixτis constructed by analogy to (4.5), namely τ=

∑

i: vertices of K

max(∇uf(u^h)∇^Ψi,0) −1

(4.10f) and againu^h is the arithmetic average of the solution over the degrees of freedom and max(A,0)is the positive part of the matrixAwhich is assumed to be diagonalisable inR with real eigenvalues.

We have left unclear the choice ofd. In practice, we choose~ _d~=~_u/||~_u||and an arbi- trary direction if~_u=0. The many experiments we have conducted shows that the non oscillatory behavior of the scheme is independent of the choice ofd. Of course for any~ direction choice will correspond a particular scheme, but all have the same non oscilla- tory behavior. The specific choice is motivated by keeping the rotational invariance of the scheme.

4.5 Boundary conditions.

We have used a simplified version of the boundary conditions that we describe for the Euler equations. If an elementK has an edge,Γ_K, on the boundary, we need to add to the degrees of freedom onΓ_Ka boundary residual. We denote it byΦ^Γ_σ^K. These residuals should satisfy the conservation relation

∑

σ∈Γ_K

Φ^Γ_σ^K=

Z

Γ_σ F_n(u^h)−f(u^h)·~_ndl

whereFnis a boundary flux. In the examples of this paper, two types of boundary are considered:

• Wall boundary conditions. The condition~_u·~_n=0 is weakly imposed so that

Fn(u^h) =





 0 p(u^h)nx

p(u^h)ny

0







• Inflow/outflow boundary conditions. The state at infinity isU∞ and we take here the modified Steger-Warming flux

Fn(u^h) = A(u^h)·~_n⁺_u^h+ A(u^h)·~_n⁺_u∞.

By analogy with what is done in [1], we have chosen a ’centered’ version of the boundary residuals, namely

Φ^Γ_σ^K=

Z

Γ_K Fn(u^h)−f(u^h)·~_nϕσ(x)dl

where again ϕσ is the Lagrange basis function defined inK forσ. This is approximated by a quadrature formula with positive weights. The quadrature formula should be of orderk+d−1, i.e. 3 for a third order scheme in 2D. The actual residual is

Φ^Γ_σ^K=|^ΓK|

∑

quadrature points

ω_quad Fn(u^h)−f(u^h)(x_quad)·~_{n. (4.11)}

In the case of interest (P²/Q² interpolation), we approximate these relation with Simpson’s formula: only one term appears in the sum and it corresponds toσ.

All the meshes we have used are made of triangles or quadrangles. We have used two type of boundary representation. In the first one we adopt a piecewise linear rep- resentation of the boundary but we might be quite far from the true geometry. In the second representation, we use a quadratic representation of the geometry. In principle, the situation should be better, but one has to be aware of two difficulties. First, the “nu- merical” representation of the boundary is notC¹in general, even if the boundary isC^∞. An example is provided on figure 2 where we approximate the boundary of a NACA012 airfoil near the symmetry axis. The second problem is that even very simple geometries, such as circle, will not be represented exactly.

0 0,01 0,02 0,03 0,04

0 0,01 0,02 0,03

Control points P2 representation Exact

Figure 2: Comparison with the true geometry between the two boundary representation methods used in this paper. The degrees of freedom are represented by circles.

The second drawback could be solved by using NURBS representation of the bound- ary, see section 7, the first one is here solved as follows: instead of trying to interpolate exactly in each boundary segment the boundary curve, we use a B´ezier representation which amounts to interpolate at the boundary points and respect the tangents at these points. We get an approximate quadratic representation of the boundary. This is the method we have used in practice.

In order to simplify the coding, we have used use an isoparametric representation of each element, even for the interior elements. The filtering operator is the adapted to this context : we need a exact evaluation of the gradient and divergence operators.

5 Relationships between Discontinuous Galerkin, stabilized Con- tinuous Galerkin and RD schemes.

We show that another interpretation of (2.3) is a variational one and then generalize it in the multi dimensional case leading to another interpretation of RD schemes that bares some similarities with DG ones.

5.1 A variational formulation of the finite volume schemes.

We start again from (2.1) in the steady case, even though the solutions are trivial. Again, we neglect the boundary conditions for the sake of simplicity, and we assume the solution to be scalar.

Formally, we introduce

β⁻_i+1/2= ^Fi+1/2−f(ui)

f(ui+1)−f(ui) ^andβ

+

i+1/2= ^f(ui+1)−Fi+1/2

f(ui+1)−f(ui)^, so that (2.3) becomes (in the steady case)

β⁻_i+1/2(f(ui+1)−f(ui))+β⁺_i−1/2(f(ui+1)−f(ui)) =0 Ifϕdiscontinuous and linear in each interval[xj,xj+1]^∗, we have

∑

j

{β⁻_j+1/2ϕj+β⁺_j+1/2ϕj+1}(f(uj+1)−f(uj)) =0. (5.1) We introduce the mappingπ^∆xfromV^∆xtoVe^∆xthe set of functions that are constant on each interval[xj,xj+1]but possibly discontinuous across the cell interfaces. The function π^∆x(ϕ)is, on[xj,xj+1],

π^∆x(ϕ) =β⁻_j+1/2ϕi+β⁺_j+1/2ϕj+1.

The relation (5.1) can be reinterpreted, sinceπ^∆xis constant on[xj,xj+1], as

∑

j

_Z

∂[x_j,x_j+1]π^∆x(ϕ)f(u^∆x)dx−

Z

[x_j,x_j+1](π∆x(ϕ))^′f(u^∆x)dx

=0. (5.2)

The next natural question is to understand how can be constructed a set of coefficients β^±_j+1/2 such that the scheme defined by (5.2) is stable, non oscillatory and accurate. The one dimensional case, for second order accuracy, is very specific. Indeed, using the tech- niques presented in section 3 and more specifically when we wish to enforce automati- cally an non oscillatory behavior, we can show that in generalβ⁻_j+1/2=0 and β⁺_j+1/2=1:

in a way, upwinding is built-in.

5.2 Generalization to the multi dimensional case.

Inspired by the variational formulation (5.2), we can generalize it and show connections of the method to more standard ones like the stream-line diffusion method or the Dis- continuous Galerkin methods by playing with the couple test function/approximation space. Again we are given a triangulationT^h, we assume that it is conformal, and denote byKa generic element ofT^h.

∗V^∆xis the set of these functions.