Toward a Higher Order Unsteady Finite Volume Solver

Based on Reproducing Kernel Methods

Sofiane Khelladia,∗, Xesus Nogueirab, Farid Bakira, Ignasi Colominasb

aLaboratoire de Dynamique des Fluides, Arts et Metiers ParisTech, Paris, FrancebDepartment of Applied Mathematics, Universidade da Coruna Campus de Elvina, A

Coruna, Spain

Abstract

During the last decades, research efforts are headed to develop high ordermethods on CFD and CAA to reach most industrial applications (complexgeometries) which need, in most cases, unstructured grids. Today, higher-order methods dealing with unstructured grids remain in infancy state andthey are still far from the maturity of structured grids-based methods whensolving unsteady cases. From this point of view, the development of higherorder methods for unstructured grids become indispensable. The finite vol-ume method seems to be a good candidate, but unfortunately it is difficultto achieve space flux derivation schemes with very high order of accuracy forunsteady cases. In this paper we propose, a high order finite volume methodbased on Moving Least Squares approximations for unstructured grids thatis able to reach an arbitrary order of accuracy on unsteady cases. In order toensure high orders of accuracy, two strategies were explored independently:(1) a zero-mean variables reconstruction to enforce the mean order at thetime derivative and (2) a pseudo mass matrix formulation to preserve theresiduals order.

Keywords: high order finite volumes, reproducing kernels, moving leastsquare approximations, mass matrix, zero-mean reconstruction

∗Corresponding authorEmail address: sofiane.khelladi@ensam.eu (Sofiane Khelladi)

Preprint submitted to Elsevier April 18, 2011

1. Introduction

The most successful numerical schemes in computational fluid dynamics(CFD) or aeroacoustics (CAA), have been spectral methods or finite differ-ences [1, 2]. These methods have proved that they are very well adapted forstructured grids, but unfortunately they present weaknesses when applied toproblems dealing with complex geometries, despite their ability to achievehigh orders of accuracy.

In this context, it is very challenging to develop methods that can solveCFD and CAA problems on unstructured grids. The finite volume method,widely and successfully used for the simulation of aerodynamics with unstruc-tured grids, seems to be a very good candidate. Unfortunately, it presentsdifficulties when it is applied to unsteady cases using DNS/LES or solvingCAA problems if we consider its most usual formulation, due to the lackof resolution of the scheme (generally limited to the 2nd order). However,rising the order of finite volume methods is not the only nor probably thebest way to improve the accuracy of the schemes, but it remains the mostusual approach on unstructured grids. This approach is also not straightfor-ward, in the sense where, the main problem becomes the evaluation of highorder derivatives and the order preservation for unsteady cases. To overcomethis difficulty, many strategies can be envisaged, see for example references[3, 4, 5, 6, 7, 8, 9, 10, 11] for more details. On the other hand, ENO andWENO schemes use adaptive stencils in the reconstruction procedure basedon the local smoothness of the numerical solution to achieve high accuracyavoiding oscillations near discontinuities. ENO schemes [12, 13] use one sten-cil among all candidate stencils in each time step. WENO schemes [14] usea linear combination of the candidate stencils to obtain a higher-order ap-proximation. Extension of ENO and WENO schemes to unstructured gridshas been presented in [15, 16, 17, 18].

In this paper, we present a finite volume method with a flux reconstruc-tion based on Moving Least Squares reproducing kernel approximations (FV-MLS). The MLS method, with its basic formulation, was first designed fordata processing and surfaces generation [19], and then, it has become verypopular in meshless and SPH methods to reconstruct related variables in asupport using a cloud of particles. Recently, this method has been appliedsuccessfully to solve steady and unsteady problems of fluid dynamics (Navier-

2

Stokes equations) on unstructured grids [20, 21]. For steady problems, fourthorder of accuracy has been achieved, whereas for unsteady problems, thirdorder of accuracy can be achieved with the addition of correction terms [22].A first approach of using FV-MLS for aeroacoustics has been presented byKhelladi et al. [23] with triangular meshes and Nogueira et al. [24] withquadrilateral meshes, both by adding correction terms to the reconstructionin order enforce the mean and to ensure the required space order of accuracy,reaching third order of accuracy.

In practice, as a Godunov scheme, FV-MLS method deals with averagevalues of the flow/acoustic field variables, and the solution is assumed to beconstant within each control volume (triangular cells). So, the spatial rep-resentation would be that of a discrete constant flow field. By analogy tothe finite element method, high order schemes are obtained by substitutingthe constant representation by a piecewise continuous ”shape function” re-construction of the flow variables inside each cell, using cell-averaged valuesfrom neighbour centroids (which constitute the stencil of the active cell). Inaddition, the proposed method allows a reconstruction of the fluxes at thecells interfaces, by the use of compact stencils and without introducing newdegrees of freedom, which is a real advantage over the most popular existinghigh-order methods.

In the present work, FV-MLS is generalized and adapted to reach higherorders of accuracy (≥ 3) for unsteady cases, by using two strategies testedindependently to enforce the mean at the time-dependent terms: (1) a zero-mean reconstruction technique and (2) a pseudo mass matrix formulationusing, respectively explicit and implicit time integration schemes.

It is obvious that the use of a mass matrix formulation is a severe com-putational burden for non-stiff problems, since explicit time-stepping wouldbe significantly more efficient. This is true if the mean value of the variablesat the time-dependent terms conserves the spacial order of accuracy. Wewill demonstrate through this paper that achieving very high order of accu-racy solving a system of equations with the mass matrix at each time step iswidely justified.Related to the previous point, it is clear that the perfect niche for the presentmethod seems to be stiff flow problems, where explicit time-stepping is notcompetitive (e.g. boundary layers). In those cases, the mass matrix does not

3

introduce any additional computational cost, and thus the advantages of theproposed higher order scheme should be more clear.

To illustrate the potentials of the proposed methodology we focus ourattention on the resolution of the linearized Euler equations (LEE) for CAAas a non-stiff problem and the compressible Navier-Stokes equations as stiffproblem for CFD. LEE are solved with aim of showing the accuracy of theproposed method. The issue of non-reflecting boundary condition is also ad-dressed with a new approach based on MLS.

The paper will be organized as follow, in the 2nd section we will presentgeneralities about the systems to be solved and higher order finite volumemethod. Section 3 will be devoted to the presentation of MLS method and tothe higher-order preserving strategies. The computational and implementa-tion aspects are discussed in section 4. Validations and examples are exposedin section 5.

2. General framework

The Navier-Stokes equations in 2D

The compressible Navier-Stokes equations for two-dimensional flow, writ-ten in cartesian coordinates and in the absence of source terms, can be castin conservative form as.

∂U

∂t+

∂

∂x(FX(U)− FV

X(U)) +∂

∂y(FY (U)− FV

Y (U)) = 0, (1)

being,

U =

uvE

.

, FX(U) =

uu2 + puv

(E + p)u

, FY (U) =

uuv

v2 + p(E + p)v

.

the conserved variables and inviscid fluxes, respectively, and

FVX(U) =

0τxxτxy

uτxx + vτxy − qx

, FV

Y (U) =

0τxyτyy

uτxy + vτyy − qy

.

4

the viscous fluxes. In the above expressions, denotes density, p pressureand uuu = (u; v) is the velocity vector.

The viscous stresses are modeled as,

τxx = 2µ∂u

∂x−

2

3

(∂u

∂x+

∂v

∂y

)

τyy = 2µ∂v

∂y−

2

3

(∂u

∂x+

∂v

∂y

)

τxy = µ

(∂u

∂y+

∂v

∂x

)

where µ is the viscosity. The heat fluxes are assumed to be represented byFouriers law

qx = −λ∂T

∂x, qy = −λ

∂T

∂y

where T denotes temperature, λ = Cpµ/Pr is the thermal conductivity, Cp

the specific heat at constant temperature (Cp = 1003.5 for air) and Pr is thePrandtl number (Pr = 0.72 for air). The equation of state and temperaturefor an ideal gas can be written as

p = (γ − 1)

(E −

1

2(u2 + v2)

), T =

1

Cv

p

(γ − 1)

where Cv is the specific heat at constant volume (Cv = 716.5 for air) andγ = Cp/Cv is the ratio of specific heats (γ = 1.4 for air). The speed of soundis given by

c =√

γp/

and the dynamic viscosity µ is assumed to be related to the temperatureaccording to Sutherlands law.

The linearized Euler equations in 2D

In many applications (as in most of aeroacoustic problems that will betreated in the numerical simulations), problems are linear, so it is possibleto linearize the Euler equations around a (mean) stationary solution U =(ρ, u, v, p). The perturbation quantities

ρ = − ρ, (u, v) = (u, v)− (u, v), p = p− p,

5

gives rise to the definition of a perturbation field

U =

ρρuρvp

which satisfies the linear conservation law with source term:

∂U

∂t+

∂

∂x(FX(U)) +

∂

∂y(FY (U)) +H(U) = S(U) (2)

with the linearized non-autonomous partial fluxes defined by:

FX(U) =

ρ u+ ρ up + ρ u uρ u v

u p+ γ p u

FY (U) =

ρ v + ρ uρ v u

p+ ρ v vv p+ γ p v

(3)

H(U) =

0

(ρ u+ u ρ)∂u

∂x+ (ρ v + v ρ)

∂u

∂y

(ρu+ u ρ)∂v

∂x+ (ρ v + v ρ)

∂v

∂y

(γ − 1) p(∂u

∂x+

∂v

∂y)− (γ − 1) (u

∂p

∂x+ v

∂p

∂y)

(4)

Where S is the source term. In case of a uniform mean flow, H is null.

2.1. Finite volume method formulation

A generalized Godunov’s scheme is a finite volume method which consistson discretizing the computational domain into a set of non-overlapping con-trol volumes in which the conservation equations are solved. The main ideaarising from the Godunov’s method [25] is to solve analytically Riemann’sproblem of the flow variables at each cell interface. The resulting solution iscell-averaged before the following time step. This is a first order method.

One way of achieving a high-order formulation of the FVM is the use ofreconstructed variables of a given order of accuracy, into the numerical flux,to achieve the desired order of the scheme. In other terms, the FVM solverwill deal with average values of the field variables, and the solution is assumed

6

constant within each control volume. So, the underlying spatial representa-tion would be that of a piecewise constant flow field. High order schemes areconstructed by substituting the constant representation by a piecewise con-tinuous reconstruction of the flow variables inside each cell. The estimationof the numerical flux, regardless of their order without reconstruction, usingthe piecewise continuous reconstruction will ensure the sought order.

To improve the order of accuracy we will use a high order reconstructionof U at the interfaces, rather than the use of centroid values. Thus, the fluxfunction is computed at the integration points at cell interfaces by means ofa Taylor expansion.

In 2D, the Taylor expansion of a regular function U at the vicinity of (I),where (I) makes reference to the centroid cell, is given by:

Ux(I)kp

(·,x) =+∞∑

α=0

α∑

β=0

1

β!(α− β)!(x− xI)

α−β(y − yI)β ∂αU

∂x(α−β)∂yβ(·,xI) (5)

The Taylor approximation is performed at the interior of each cell, toestimate the value of the variable at the integration point x

(I)kp at the edge

of the cell I. On unstructured grids, the main difficulty of evaluating Ux(I)kp

is

the estimation of higher order derivatives. The FV-MLS method overcomesthis difficulty by using Moving Least Approximations [19] to compute thederivatives. This approach is somewhat different that the usual approachof high-order finite volume schemes. Thus, the usual approach is pragmaticand bottom-up. Starting from an underlying piecewise constant represen-tation, a discontinuous reconstruction of the field variables is performed atthe cell level. We start from a high-order and highly regular representationof the solution, obtained by means of Moving Least-Squares approximation,and well suited for general, unstructured grids. This approach is directlysuitable for the discretization of elliptic/parabolic equations and high orderspatial terms [21]. For equations with a predominantly hyperbolic character,the global representation is broken locally, at the cell level, into a piecewisepolynomial reconstruction, which allows to use the powerful finite volumetechnology of Godunov-type schemes for hyperbolic problems (e.g. Riemannsolvers, limiters).

7

Note that this approach is made for the convective terms of the equations.Viscous fluxes and source terms are approximated directly at integrationpoints by using MLS. This is a distinctive feature of the proposed scheme.It allows a clear and centred high-order discretization of viscous terms. Foran extensive explanation about the treatment of viscous terms, we refer theinterested reader to [21, 26].

Consider a cell-centred finite volume approach with triangular cells in 2Dand a flux that take the formFFF = (FX ,FY ) for LEE orFFF =

(FX − FV

X ,FY − FVY

)

for Navier-Stokes equations.The final semi-discrete scheme for the system to be solved can be written as

∫

Vi

∂U

∂tdV = R(U)

where R(U) is the residual of the system given by,

R(U) = −

∮

Γi

FFF(U) · ndΓ +

∮

Vi

S(U)dV

3. Finite volume methods based on MLS reconstruction

In this section, a succinct presentation of MLS reconstruction applied tohigher-order finite volume method is presented. A large part of the study thatfollows is devoted to the construction, for the first time with FV-MLS, of aspecific mass matrix formulation who preserves the spatial order of accuracyof unsteady stiff and non-stiff problems. An implicit time solver is setup tosolve the full mass matrix problem. A zero-mean reconstruction formulationusing an explicit time solver is also presented.

3.1. MLS local reconstruction

The Moving Least Squares approximations [19] is a very usual techniquein the meshless community. It is very well suited for the approximationof scattered data. In this section we introduce briefly this technique. Werefer the reader to [19, 27, 20, 21] and references therein. Let us considera function Φ : Rd → RN defined in a domain Ω. The basic idea of theMLS approach is to approximate Φ in the vicinity of a given point taken asreference, through a weighted least-squares fitting procedure. In the following

8

we consider the approximation of Φ in the vicinity of the centroid. Note thatthe approximation in the vicinity of any other point would follow the sameprocedure.

Definition 1 (Mesh and stencil). Let us consider a finite volume mesh.Let us consider cell with a centroid xI . Let hxI

> 0 and let us define a stencil

S(I) := j ∈ N, xj ∈ B(xI , hxI), card(S(I)) = nxI

.

Here, B(xI , hxI) denotes the ball centred at xI with a radius hxI

. Inpractice, the real number hxI

can always be chosen large enough in order todefine a stencil with a prescribed ”number of neighbours” nxI

.

Definition 2 (Kernel function). Let W[xI ] ∈ C0(Rd) be a function withcompact support B(xI , hxI

) satisfying

∀j ∈ S(I), W[xI ](xj) > 0.

Definition 3 (Local MLS reconstruction function). Let xI ∈ Rd be thecentroid of a cell and S(I) its associated stencil. For any function Φ ∈

C0(Rd), we denote by Φ[xI ] ∈ C0(Rd) the function defined by

∀x ∈ Rd, Φ[xI ] (x) := pT (x) ·A[Φ,xI ] (6)

where pT : Rd 7→ M1,m(R) is an m-dimensional polynomial basis (withm ≤ nxI

) and A[Φ,xI ] ∈ Mm,N(R) is a set of parameters which minimizethe following quadratic functional:

J[Φ,xI] (A) =

nxI∑

j=1

W[xI ](xj)∥∥Φ(xj)− pT (xj) ·A

∥∥2 .

Remark 1. It means that Φ[xI ] is the best approximation of Φ in the vicinityof xI in the sense of the mean least squares, weighted by the kernel W[xI ].

For example, a polynomial cubic basis is given by:

p(x) =(1 x y xy x2 y2 x2y xy2 x3 y3

)T(7)

9

which provides cubic completeness. In the above expression, (x, y) denotesthe Cartesian coordinates of x. In order to improve the conditioning, thepolynomial basis is locally defined and scaled: if the shape functions aregoing to be evaluated at xI , the polynomial basis is evaluated at (x−xI)/h,where h is the smoothing length, which is a measure of the size of the supportΩxI

.

Proposition 1 (Interpolation structure of the MLS function [20, 21]).The MLS reconstruction procedure admits the following representation:

U[xI ](x) = N[xI ](x)T · UΩxI

where vector UΩxIcontains the values of U at each centroid of the stencil, as

UΩxI:=

U(x1)...

U(xnxI)

∈ MnxI

,N(R). (8)

and the basis functions are given by

N[xI ](x)T = p

(x− xI

h

)T

·(PΩxI

·W[xI ](x) · PTΩxI

)−1

· PΩxI·W[xI ](x) (9)

where the matrices PΩxIand W[xI ](x) read as follows:

PΩxI=

(p

(x1 − xI

h

)· · · p

(xnxI

− xI

h

))∈ Mm,nxI

(R),

W[xI ](x) = diag(W[xi](x)

)i=1,...,nxI

∈ MnxI,nxI

(R).

Corollary 1 (Interpolation structure of the MLS function [20, 21]).The local MLS reconstruction function is generated by the family

SpanN

(j)[xI ]

j=1,...,nxI

.

withNT

[xI ]:=(N

(1)[xI ]

. . . N(nxI

)

[xI ]

).

10

Thus, any MLS reconstruction function U[xI ] can be identified by its local

MLS coordinates Uj := U(xj) ∈ RN related to the basis N(j)[xI ]

j=1,...,nxI, i.e.

U[xI ](x) =

nxI∑

j=1

N(j)[xI ]

(x)Uj. (10)

We remark that the approximation is written in terms of the MLS ”shapefunctions” N[xI ](x)

T , that are depending on the grid. So, in fixed grids theywill be computed only once. The variables and its derivatives are computedby using these shape functions. Thus, in the FV-MLS method, the MLSapproximation will be used to compute the derivatives needed for the recon-struction of variables at quadrature points at cell interfaces. We rememberthat the reconstruction step is made by using a Taylor expansion until orderN as presented in the previous section.

Moving Least-Squares Reproducing Kernel Approximations (MLS) pre-sented in the previous section, is used to construct the local high-order con-tinuous representation of Φ and then its space derivatives. What we needto retain is that the general local representation of Φ (in the vicinity ofxI), which is approximated or reconstructed, is sought for in the subspacespanned by a set of basis functions

SpanN

(j)[xI ]

j=1,...,nxI

associated to the nodes, such that Φ(x) is locally represented by a functionof the form:

Φ[xI ](x) =

nxI∑

j=1

N(j)[xI ]

(x)Φj (11)

in the vicinity of xI . Eq. (11) states that the approximation at a point x inthe vicinity of xI , is computed using nxI

surrounding nodes.

Remark 2. The definitions above can be generalized to any reference pointother than the cell centroid. In particular, it is possible to take integrationpoints as reference points. As we have commented in section 2, we use MLSreconstruction for the viscous fluxes directly at integration points.

11

Let us come back to Eq. (5). Now all the derivatives can be computedusing Eq. (11) as

∂αΦ

∂x(α−β)∂yβ(xI) ≈

nxI∑

j=1

∂αN(j)[xI ]

∂x(α−β)∂yβ(xI)Φj (12)

Since N(j)[xI ]

being only related to the grid topology, it is important to notice

that for a fixed grid (non-deforming grid), the shape function and its deriva-tives are calculated just once at the beginning of computation so the increaseof the computational cost is low.

In a triangular unstructured grid, if we consider xI the centroid of theactive cell, then j are the centroids of the nxI

surrounding cells. This leadsto the construction of a stencil around xI (see Fig. (3.1)).

bc

bcbc

bc

bc

bc

bcbc

bc

bc bc

bc

bc

bcbc

bc

bc

bcb

rsrs

rsrs

rs

rs

rsrs rs

rs

rs

rs

rsrs

rsrs

rs

rs

ΩX

rs

b IJ

Figure 1: Stencil support

The number and distribution of cells around xI in a stencil will be dis-cussed below.

Kernel functions

The kernel function plays a key role in the properties of the scheme [28].Many functions may be used as kernels, as splines, Gaussians, exponentials.We refer the reader to the book of Liu [29] for more details.

An example of kernel functions is the 1D exponential kernel function,which is very well suited for anisotropic node distribution:

W (xj − xI , κ) =e−(

sc)

2

− e−(dmc )

2

1− e−(dmc )

2 (13)

12

with s = |xj − xI | with j = 1, . . . , nx, dm = max (|x− xI |), c = dm2κ, xI

is the reference point (the point around which the stencil moves, in thiscase the centroid of each cell, I), and κ is a shape parameter. A 2D kernel isobtained by multiplying two 1D kernels. κ varies from 1 to 6 according to theapplications. A complete analysis of the influence of the kernel parameterson the properties of the numerical scheme can be found in [28]. Here, weshow the dispersion and dissipation characteristics of the FV-MLS 3rd orderscheme for different values of κ.

Figure 2: Dispersion (left) and dissipation (right) curves of the 3rd order scheme FV-MLSmethod for different values of the shape parameter of the exponential kernel κx.

In Fig. 2 it is shown that the dispersion and dissipation characteristics ofthe FV-MLS method are strongly related to the choice of the κ parameter.For Navier-Stokes or LEE, a value of κ between 2.5 and 3.0 seems to be a goodchoice. It guarantees the good conditioning of the momentum matrix PΩxI

·

W[xI ](x) · PTΩxI

and an acceptable compromise of dispersion and dissipation.

3.2. Time-dependent formulation

Let us show how the reconstruction (Eq. (5)) induces a formulation basedon a differential system with a non-diagonal sparse mass matrix (M = [µij ])for unsteady cases. In general, the diagonal structure is recovered by enforc-ing reconstructions that preserve the mean values. But enforcing µij = 0 fori 6= j induces a loss of accuracy of the reconstructed variables (see Barth[4, 5, 30] and Venkatakrishnan [31, 32] for more details).

13

Construction of the time derivative contribution

In the vicinity of a volume control with centroid xi, according to Eq. (11),x → U(·,x) is locally approximated by

U(·,x) ≈∑

j∈S(i)

N(j)[xi]

(x)Uj . (14)

where S(i) denotes the so-called stencil of the MLS reconstruction in thevicinity of xi. More precisely, we have

S(i) = j ∈ N, xj ∈ Ωxi, nxi

= card(S(i)).

Let us replace Eq. (14) in the volume integration of the time derivative:

∫

Vi

∂U

∂t(·,x) dx ≈

∫

Vi

∂

∂t

∑

j∈S(i)

N(j)[xi]

(x)Uj

dx

=d

dt

∑

j∈S(i)

(∫

Vi

N(j)[xi]

(x) dx

)Uj

.

For a fixed grid, N(j)[xi]

is independent from time, then,

∫

Vi

∂U

∂t(·,x) dx ≈

∑

j∈S(i)

(∫

Vi

N(j)[xi]

(x) dx

)dUj

dt.

This computation leads us to define the pseudo-mass matrix by

M = [µij] .

with

µij =

1

meas(Vi)

∫

Vi

N(j)[xi]

(x) dx, if j ∈ S(i),

0, otherwise,(15)

such that ∫

Vi

∂U

∂t(·,x) dx ≈ meas(Vi)

∑

j∈S(i)

µij

dUj

dt. (16)

14

Remark 3. Recall that the classical low order representation (based on thepiecewise constant representation of function U) yields∫

Vi

∂U

∂t(·,x) dx ≈ meas(Vi)

dUi

dt, with Ui =

1

meas(Vi)

∫

Vi

U(·,x) dx.

It can be seen as a low-level representation defined by Eq. (16) (with a polyno-mial degree equal to 0) leading to the property µij = δij (Kronecker symbol).Thus the coefficients µij can be seen as weight parameters whose distributionis highly related to the required precision, taking into account the reconstruc-tion of the derivatives.

Remark 4 (Consistency of the mass matrix formulation). The MLSshape functions have some particular properties. In particular, the MLS re-construction preserves the constant states, i.e.

∀x ∈ Ωxi,

∑

j∈S(i)

N(j)[xi]

(x) = 1

so that, if we define µdi =∑

j∈S(i)

µij, then the following property holds:

µdi =∑

j∈S(i)

µij =

nxi∑

j=1

1

meas(Vi)

∫

Vi

N(j)[xi]

(x) dx

=1

meas(Vi)

∫

Vi

∑

j∈S(i)

N(j)[xi]

(x)

︸ ︷︷ ︸=1

dx = 1. (17)

Eq. (17) can be considered as a property of mass matrices obtained by MLS-based shape functions. It can be an excellent consistency test for Eq. (15) asthe coefficients are computed numerically (eventually using approximationsas will be stated in the forthcoming description).

Let us explain the numerical procedure leading to the effective computa-tions of the coefficients µij . As stated by Eq. (15), the initial mass matrix

elements are obtained by the integral of N(j)[xi]

for each neighbor and over each

cell. Thus, we use (i) a Taylor expansion of the shape function and (ii) anaproximation of the space derivatives of the shape function at the centroid:

15

(i) Taylor expansion. One can approximate this integral for a given orderby making a Taylor expansion of the shape function:

N(j)[xi]

(x) =+∞∑

α=0

α∑

β=0

1

β!(α− β)!(x− xI)

α−β(y − yI)β

∂αN(j)[xi]

∂x(α−β)∂yβ(xI).

Using this formula with Eq. (15) gives

µij =1

meas(Vi)

+∞∑

α=0

α∑

β=0

1

β!(α− β)!

(i)α−β,β

∂αN(j)[xi]

∂xα−β∂yβ(xi) (18)

with

(i)α,β =

∫

Vi

(x− xi)α(y − yi)

β dx dy.

which can be estimated using a symmetric quadrature presented in [33]by Wandzura et al. The infinite upper bound of the sum in Eq. (18) isreplaced by a finite one for following formulations.

(ii) Estimation of the shape function derivatives. Recalling Eq. (10) theshape functions are given by

N[xI ](x) = p

(x− xI

h

)T

· C[xI ](x),

where the momentum matrix C[xI ] : Rd 7→ Mm,nxI

(R) is defined by

C[xI ](x) =(PΩxI

·W[xI ](x) · PTΩxI

)−1

· PΩxI·W[xI ](x).

Higher order derivative of N[xI ](xI) at the centroid xI is then given by:

∂αN[xI ]

∂x(α−β)∂yβ(xI) =

∂αp

∂x(α−β)∂yβ(0) · C[xI ](xI)

+p(0) ·∂α

C[xI ]

∂x(α−β)∂yβ(xI). (19)

In Eq. (19), the derivation of the polynomial p is straightforward, unlikethe derivation of C[xI ] which is difficult to estimate analytically forderivatives of degree greater than 2. This is why we assume that forα > 2,

∂αN[xI ]

∂x(α−β)∂yβ(xI) ≈

∂αp

∂x(α−β)∂yβ(0) ·C[xI ](xI). (20)

More details about the procedure used in this paper can be found in[21, 22].

16

Construction of the residual

The residual ratio includes the flux divergence term and the source term.

Let us recall that the flux divergence term is approximated by a quadra-ture formula and a numerical flux. In order to take into account the high-order reconstruction of the function in the numerical flux evaluated at theinterfaces of the control volumes, we use the Taylor reconstruction functionsfor the left/right states:∮

∂Vi

FFF(t,U) · n ≈

Ne(i)∑

k=1

Ng(i|k)∑

p=1

(H(t,x

(i)kp ,U

+(t,x(i)kp ),U

−(t,x(i)kp ),n

(i)kp

)−HV (U[xkp])

)ℓ(i)kp W

(i)kp ,

(21)

where Ne(i) denotes the number of edges of the control volume with cen-troid i, Ng(i|k) is the number of Gauss quadrature points on the k-th local

edge of the control volume with centroid i and x(i)kp denotes the p-th Gauss

quadrature point on the k-th local edge of the control volume with centroid i.H and HV are respectively the numerical flux that corresponds to the con-vective part and the viscous part of flux F defined above.

The left and right states of the variables U+ and U− have still to beprecised:

• the left state is defined by the Taylor reconstruction function relatedto the active cell (with centroid i), i.e.

U+ = Ux(i)kp

(t,xi)

• the right state is defined by the Taylor reconstruction function relatedto centroid j

U− = Ux(i)kp

(t,xj)

such that cells i and j have the local edge k in common.

The source term∫ViS(t, ·,U(t, ·)) is evaluated by quadrature formulas

leading to the definition of a discrete source term

meas(Vi) Si

17

where Si is approximated directly at the quadrature points by using MLSreconstruction.

Thus the flux divergence and sources can be grouped in a residual functionapproximated by

Ri(·,U, U) = −

Ne(i)∑

k=1

Ng(i|k)∑

p=1

(H(t,x

(i)kp ,U

+(t,x(i)kp ),U

−(t,x(i)kp ),n

(i)kp

)−HV (U[xkp])

)ℓ(i)kp W

(i)kp

+meas(Vi) Si

In this paper, we use the Roe approximate Riemann solver [34].

Remark 5. One of the major advantages of the proposed method is the use ofMLS approximants as a global reconstruction procedure to evaluate the viscousfluxes at the quadrature points on the cell edges. This procedure providesalso an accurate high-order flux and a straightforward discretization of theseterms. See [21] for more details.

Resulting system to solve

To recover the orders and enforce the accuracy of the scheme we shouldsolve, after discretization in space, the following system of coupled ODE:

dV ·M ·U

dt= R(·,U)

withV := diag(meas(Vi)).

The pseudo-mass matrix expresses the relationship between the vertex rep-resenting the control volume and its nearest neighbors. M is only a functionof the grid and hence a constant mass matrix for a static grid. Redefiningthe source term R(·,U) = V

−1 · R(·,U) the system to be solved reads, then,

M ·dU

dt= R(·,U). (22)

Matrix M is generally sparse and can be very large, see Fig. (3).

18

0 200 400 600 800 1000 1200 1400

0

200

400

600

800

1000

1200

1400

Number of cells

Num

ber

of c

ells

Figure 3: 4th order scheme mass matrix for one variable before assembling.

3.3. Time integration: explicit solver (zero-mean method)

A sine qua non condition to enforce the conservation of the mean is torecover the “diagonality” of the mass matrix by replacing the continuousreconstruction in the time-dependent term by the discontinuous one U. Atfirst sight, this method may appear as a mass lumping procedure, the differ-ence lies in the manner of estimating the diagonal elements of M in functionof columns or lines (depending on the system representation). A manner toenforce the conservation of the mean is the use of zero-mean polynomials inEq. (5). In this case Eq. (5) reads

UI(·,x) = U(·,xI) + (x− xI)∂U

∂x(·,xI) + (y − yI)

∂U

∂y(·,xI)

+

+∞∑

α=2

α∑

β=0

[(x− xI)

(α−β)(y − yI)β −Θαβ

]

β!(α− β)!

∂αU

∂x(α−β)∂yβ(·,xI)(23)

whereΘαβ =

x(α−β)yβ

meas(VI)

19

is the correction term ensuring the zero-mean, and

xαyβ =

∫

Vi

(x− xi)α(y − yi)

β dx dy

is estimated using a symmetric quadrature presented in [33] by Wandzuraet al. With this procedure, the consistency between the resulting diagonalmass matrix and the residual term, R(·,U), is ensured by the use of the “bro-ken” reconstruction given by Eq. (23). This reconstruction can be found insome mean-conservation restrictions used in other cell averaged finite volumemethod schemes, see [4, 35] for instance. In our case, the mean-conservationcorrection appears a posteriori in the reconstruction contrary to the k−exact[5] method for example, in which we impose a priori mean-conservation re-striction. The system to be solved becomes then

dUI

dt= RI(·, U). (24)

The above system can be solved using explicit time schemes. The mostsuccessful ones in CAA are Runge-Kutta schemes family, particularly thoseensuring low-dissipation and low-dispersion time integration. In this context,a low-storage fourth-order six-stage Runge-Kutta scheme based on the initialWilliamson’s formulation [36], is used. The variant of Williamson’s formu-lation considered here is that optimized by Berland et al. [37] (RK46-NL)ensuring the fourth order of accuracy in nonlinear. The scheme seems to bethe most appropriate for our case since it presents a large stability range.Denoting Un the numerical solution evaluated at time tn and let us definethe time-step ∆n = tn+1 − tn. The algorithm reads as follows:

1. Initialization:U

(0)I = Un

I , Θ(0)I = 0.

2. RK loop: for i = 1, ..., 6

Θ(i)I = aiΘ

(i−1)I +

∆n

meas(VI)RI(tn + ci∆n, U

(i−1)) (25)

U(i)I = U

(i−1)I + biΘ

(i)I .

3. Update:Un+1

I = U(6)I .

20

where ai, bi and ci are the optimized coefficients of RK46-NL schemegiven below in table 1.

It is important to note that the residual U in Eq. (25) is estimated usingthe zero-mean reconstructed variable and computed at (i − 1)th sub-timestage.

i ai bi ci1 0.0 0.032918605146 0.02 0.737101392796 0.823256998200 0.0329186051463 1.634740794341 0.381530948900 0.2493517233434 0.744739003780 0.200092213184 0.4669117050555 1.469897351522 1.718581042715 0.5820304140446 2.813971388035 0.27 0.847252983783

Table 1: Coefficients ai, bi and ci of the RK46-NL scheme.

It is well known that explicit time integration schemes remain subject toa CFL restricted time step. It may pose a problem when we are only in-terested in boundary layer problems or acoustics very far from sources. Foraeroacoustics, other strategies can be adopted, for instance, the use of hybridprocedure LEE and Integral Methods [38] on which can be very interestingfor free and far field acoustic wave propagation. But for boundary layerproblems implicit approaches are required.

3.4. Time integration: implicit solver

In practice, implicit time integration solvers are more difficult to imple-ment than explicit solvers. The reason is that they require heavy linearalgebra tools and especially more computational resources. But their lowrestriction on CFL and their ability to preserve the spatial order of accuracyby taking into account the full mass matrix (discussed above), are their ma-jor strengths.

The problem to be solved given by Eq. (22) can be considered as aninitial value problem of Differential-Algebraic-Equations (DAEs). Amongseveral solvers, more or less adapted for our case, we used IDA [39] for Im-plicit Differential Algebraic solver, a C-based programming language variant

21

of the well known DASPK solver [40, 41]. As major feature, the time integra-tion method is the variable-order (from 1 to 5), variable-coefficient BackwardDifferentiation Formula, with adaptive time-step, see [42, 43] for more details.

For the solution of the linear system (22), an inexact Newton/Krylovsubspace iterative corrections based on a scaled preconditioned GMRES [44](Generalized Minimal Residual method) solver is used. Only left precondi-tioning is allowed in IDA code. Typical preconditioners used with this codeare based on the approximation of the Jacobian matrix of the system givenby (22). In our case, the left preconditioner matrix is given by,

P = χM−∇UR(·,U),

where χ is a scalar inversely proportional to the time step. For sake ofsimplicity, we consider

P ≈ χM− diag (∇UR(·,U)) ,

accompanied by a zero fill-in incomplete LU (ILU(0)) factorization, see [45]for more details. The obtained system using P is solved via a standard LUdecomposition solver.

This approximation can appear as a “crude” approximation of the Ja-cobian matrix, but using even a poor approximation can be surprisinglysuperior to using the Newton-Krylov method with no preconditioning [45].

4. Computational aspects

4.1. Stencils: size and distribution

There exists an optimal size nxIof points in the stencil such as Nmin ≤

nxI≤ Nmax. The minimum number of points correspond to the dimension

of the polynomial basis, it is given by

Nmin := m =(d+ order)!

d! order!(26)

• for d = 2 : Nmin =1

2(order + 1)(order + 2)

• for d = 3 : Nmin =1

6(order + 1)(order + 2)(order + 3)

22

The maximum number of points in a stencil is more difficult to estimate. Itcan be defined as

Nmax = Nmin + n+ (27)

where n+ is the additional number of points to be defined. It was observedthat if p is too large, then the results present an excessive dissipation. De-pending on the orders and the quality of the grid, we take 5 ≤ n+ ≤ 15. Anoptimization study should give more information to define this parameter.

Once the size of the stencil is defined, the step after concerns the distri-bution of the particles (centroids) around the active cell. If possible, priorityis given to construct stencils as compact as possible by using layers of cellsaround the active cell, see fig. (4 A)). But in practice the fulfillment of thecompactness condition requires a high number of points in the stencil, whichinduces dissipation as mentioned above. To overcome this inconvenient, lastparticles which can not form a compact layer are placed such as satisfyinga barycentric equilibrium (see fig.(4 B))). It is a choice among others, butit is more rigorous and maybe more accurate to define a distribution cri-terion based on the singularity-avoiding [46], the best momentum matrix

conditioning(PΩxI

·W[xI ](x) · PTΩxI

)of equation (9) and/or minimum of dif-

fusion. Concerning this last point, when the dissipation is very small, thescheme can become unstable, this is due to the fact that waves with highwavenumber are not well solved by the numerical scheme and present highdispersion errors (see figure 2). This is why one should be very careful whendefining the minimum diffusion criterion.

23

A) B)

Figure 4: MLS 5th order scheme stencil (4th order polynomial basis): Nmin = 15 pointsand Nmax = Nmin + 4 = 19 points (eq. (27)). A) Compact stencil distribution, B)non-compact stencil distribution with respect to Nmax = 19 points.

4.2. Boundary conditions

For our modeling, the boundary conditions enter in the discretized equa-tions through a proper definition of the numerical flux. The numerical in-tegration at the Gauss points at the boundary interfaces can be written asH(·, ·,U+,U∗−,n), where n is the outward normal unit vector from the do-main and U∗− is the external state variable. Depending on the boundarytype, the construction of U∗− accounts for, both, the physical boundary con-ditions that must be enforced and the information leaving the domain.

In this paper, for Navier-Stokes equations we use no-slip boundary condi-tion to model rigid walls and far-field boundary condition to model the exter-nal boundaries of the computational domain. For LEE, a perfectly reflectingboundary condition is easily obtained by defining, at each Gauss points onthe rigid wall boundaries, an external mirror fictitious state UUU ∗−. For ex-ternal boundaries, we propose a new methodology to model non-reflecting(absorbing) boundary conditions based on combining a stretching zone withMLS filtering and an upwinding technique used by Bernacki et al. [47] withDG.

24

4.2.1. Absorbing boundary conditions for CAA problems

Constructing absorbing (non-reflecting) boundary conditions for CAA ispretty delicate because of the high sensitivity of the accuracy to the smallspurious wave reflexions at far field boundaries. Approaches based on thecharacteristics theory are not suited for CAA problems, other approaches,such as Perfectly Matched Layers (PML) [48] and radial boundary condition[49] are more indicated and widely discussed in the literature for finite dif-ferences schemes.

In this work we employ the upwinding technique used by Bernacki et al.[47] with DG to select only outgoing waves at the outer boundaries. Intu-itively, it means that the whole wave of energy is dissipated at boundaries.Unfortunately nothing prove that energy is actually dissipated and no spu-rious wave reflexions persist. To overcome this inconvenient, we join to theabove procedure a grid stretching zone [50, 24]. Grid stretching transfersthe energy of the wave into increasingly higher wavenumber modes and thenumerical scheme removes this high-frequency content. With this processmost of the energy of the wave is dissipated before reaching the boundaries.This becomes clear by looking at figure (2), where the dispersion and dissi-pation properties of the FV-MLS scheme are shown. These properties arerelated to the real and complex part of the numerical wavenumber. For ahigh wavenumber the numerical method introduces more dissipation.

At the grid stretching zone, it is possible to use the MLS method as afilter even in unstructured grids. The filtering process is developed by theapplication of a MLS reconstruction of the variables, i.e.

U(x) =

nxI∑

j=1

U(x)N(j)[xI ]

(x)

where, U is the reconstructed variable, U is the filtered variable and N(j)[xI ]

isthe MLS shape function. This reconstruction is performed by using a kernelwith shape parameters favoring dissipative behavior that the ones used to theapproximation of the variables. The value of these parameters determinesthe range of frequencies to be filtered.

25

At the outer boundaries, we propose the following explicit numerical flux,

H(·, ·,Un,U∗n,n) =1

2(F(·, ·,Un) · n+ |P| ·Un−1) (28)

with the following notations:

• U∗n is the fictitious state corresponding to the absorbing side ensuringthe condition P ·U∗n = |P| ·Un−1;

• P is the Jacobian matrix of system (2) and |P| = V−1 · |D| ·V;

• D and V are, respectively, eigenvalues diagonal matrix and eigenvectorsmatrix of P;

• |P| then is given by

L3nx

2c0(−L1 + L2)

ny

2c0(−L1 + L2)

−L3

c20+

L1 + L2

2c0

0n2x

2(L1 + L2) + n2

yL4nxny

2(L1 + L2 − 2L4)

nx

2c0(−L1 + L2)

0nxny

2(L1 + L2 − 2L4)

n2y

2(L1 + L2) + n2

xL4ny

2c0(−L1 + L2)

0nxc02

(−L1 + L2)nyc02

(−L1 + L2)L1 + L2

2

with

L1 =∣∣V · n− c0

∣∣ , L2 =∣∣V · n+ c0

∣∣ , L3 = L4 =∣∣V · n

∣∣

and V = (u, v) and c0 the sound velocity.

4.3. Implementation aspects

Before passing to the validation part, some technical points concerningthe memory footprint (compared to DG) and implementation aspects shouldbe discussed.

In [51] a comparison between the fourth-order DG and FV-MLS methodshas been presented for steady problems and quadrilateral grids. It has beenshown that both methods present practically the same accuracy, but theFV-MLS method uses a smaller number of degrees of freedom. Thus, for

26

the Euler equations the fourth-order FVMLS method requires a number ofdegrees of freedom that is roughly 1.78 times smaller than the p = 3 DGmethod (p is the polynomial degree). This ratio is even bigger in the caseof Navier-Stokes computations. Moreover, in [26] it is performed a compar-ison between the storage requirements of the CDG and FV-MLS numericalscheme for elliptic problems. In [51] and [26], grids were made of quadri-lateral elements. Here the stencil for the FV-MLS method is different (seefigure (4)), and we present the comparison for this situation. Thus, in orderto obtain the equivalent grid resolution, we consider a p = 4 DG triangularelement and we obtain 16 control volumes, as it shown in the figure:

Figure 5: Subdivision of a triangular p = 4 DG element to obtain 16 control volumes.Shaded points are the centroids of each control volume

Thus, with triangular grids and p = 4 DG elements, the number of de-grees of freedom (per variable) needed to obtain the same space resolution isslightly higher for the FV-MLS method (15 in the case of DG and 16 for theFV-MLS scheme). We note that this situation changes when quadrilateralgrids are used, (see [51] and [26]). In case of elliptic problems, DG per-forms an additional work to eliminate the auxiliary variables required in theformulation. For this case, it is interesting to compare the storage require-ments, by comparing the numbers of entries in the coefficient matrix [26].This comparison is made for the same grid resolution. We only comparewith the Compact Discontinuous Galerkin scheme (CDG) [52] because is themost favorable case for the DG scheme in case of elliptic terms. Recallingfigure (4) , we have 19 nodes that are part of the stencil of a given cell I, forthe fifth-order FV-MLS method. Then the scheme requires 16 × 19 = 304entries in the coefficient matrix. In the case of the CDG scheme, the total

27

number of non-zero entries is 450 [52]. This number is even higher for otherDG schemes such as LDG (Local Discontinuous Galerkin) or the Bassi-Rebayscheme (BR2) [52].

It is well known that if one wants to preserve the targeted order of accu-racy of the FV scheme, all the internal intermediate numerical integrationsor differentiations should be estimated at least with the same order of accu-racy. For these considerations, we used for the present work the followingapproximations:

• 8th order Weddle’s rule [53] integration of the flux in equation (21).

• 7th order symmetric quadrature [33] for the numerical integration usedin equations (23) and (18).

• The infinite upper bound of the sum in equation (18) is stopped at the7th order. Consequently, for the estimation of the mass matrix elements,the shape function derivatives are pushed until 7th order and then aminimum of (7 + 1) × (7 + 2)/2 = 36 cells in the stencil surroundingthe active cell are used.

These high order approximations are used for all space schemes from orders2 to 5. The idea behind this choice is to break the correlation between er-rors due to the FV scheme and those induced by the other approximations,i.e., avoid the influence of the other approximations when examining schemeerrors and slopes.(See examples and validations section)

5. Validation and examples

In this work we only present 2D examples. Note that all the above com-mented is straightforwardly applicable to 3D. In fact the zero-mean approachhas been already applied to turbulence computations [54] and to 3D test cases[22]. The methodology we presented for implicit approach using full massmatrix will not change in 3D. The additional cost of the present method isonly the computation of the MLS shape functions and its derivatives. As thisis performed only once (on fixed grids) at the beginning of the computations,

28

the additional cost is only related to the inner loops across the neighboursof a given cell to compute the gradients, following equation (12).

5.1. Order validation

To estimate the order of accuracy of a given unstructured solver it iscommon to examine the evolution of the L2 norm error versus the grid size.Knowing that the log of the L2 error varies linearly with the log of the onedimensional grid size, the slope corresponds to the order of the scheme. TheL2 norm of the pressure error and the order of convergence of the numericalsolution are given by

L2e =

√∑Nc

i=1

∫Vi(p− pex)

2dV

V(29)

order =

√√√√√√√

log

(L2error1L2error2

)

log

(√Nc2

Nc1

) (30)

Where,Nc the cells number on the grid, Vi the area of a cell, V the overall area of thedomain, p the computed pressure and pex the exact solution of the pressure.

5.1.1. Acoustic pulse problem

The objective of this test is to verify the order of accuracy of the FVscheme by comparing the analytical and the actual convergence rates usingconfigurations with full mass matrix and zero-mean reconstruction in anunstructured grid. The problem concerns the propagation of a Gaussianpulse using non-dimensional LEE where the analytical solution can be easilycalculated [49]. The mean velocities are set to zero and the computationaldomain is a square extending from −100 to 100 in the two space directions.The source is located at xs = ys = 0, and it is defined as,

S = exp

(−ln(2)

9(x2 + y2)

)× [1 0 0 1]T (31)

29

Grid L2e-o(2) order L2e-o(3) order L2e-o(4) order L2e-o(5) order2020 4.666E-03 - 4.964E-03 - 3.920E-03 - 3.613E-03 -3588 2.663E-03 1.95 2.085E-03 3.02 1.540E-03 3.25 1.462E-03 3.154548 2.093E-03 2.03 1.488E-03 2.85 1.068E-03 3.09 1.019E-03 3.048102 1.141E-03 2.10 5.823E-04 3.25 4.088E-04 3.33 3.827E-04 3.3915470 5.735E-04 2.13 2.175E-04 3.05 1.468E-04 3.17 1.401E-04 3.11

Table 2: Zero-mean reconstruction FVM convergence rate for the pressure solution att = 25 in an unstructured grid

Grid L2e-o(2) order L2e-o(3) order L2e-o(4) order L2e-o(5) order2020 5.321E-03 - 5.888E-03 - 1.914E-03 - 1.347E-03 -3588 3.064E-03 1.92 2.458E-03 3.04 5.842E-04 4.13 3.216E-04 4.994548 2.405E-03 2.04 1.740E-03 2.91 3.601E-04 4.08 1.822E-04 4.798102 1.314E-03 2.09 7.349E-04 2.99 1.171E-04 3.89 3.827E-05 5.4115470 5.989E-04 2.43 3.017E-04 2.75 3.371E-05 3.85 7.414E-06 5.08

Table 3: Full mass matrix FVM convergence rate for the pressure solution at t = 25 in anunstructured grid

The results are picked at t=25 before the wave reaches the domain bound-aries. Tables (3) and (2) presents the convergence rate of the two proposedmethods. Looking the results, which appears first is that, the maximum or-der of accuracy can reached by the zero-mean reconstruction is 3. This is duethe correction term, in equation (23), in which the second order error seemsto be predominant. The L2 norm errors corresponding to the 4th and 5thorders are smaller than those corresponding to 2nd and 3rd orders, see alsofigure (6). Now, concerning the full mass-matrix method, all the targetedorders are respected and higher-orders are more accurate. Between the twomethods, one can only compare 2nd and 3rd orders results. As shown inthe above tables and figure (6), the zero-mean reconstruction seems to bemore accurate, this is due to the time integration method. In fact, explicitRK46-NL algorithm is optimized for this kind of applications and it is moreaccurate than the implicit preconditioned Newton/Krylov GMRES solver.

30

1e-06

1e-05

0.0001

0.001

0.01

1000 10000 100000

L2 E

rror

Grid

with zero-mean, 2nd orderwith zero-mean, 3rd orderwith zero-mean, 4th orderwith zero-mean, 5th order

with mass matrix, 2nd orderwith mass matrix, 3rd orderwith mass matrix, 4th orderwith mass matrix, 5th order

Figure 6: Convergence rate of the Gaussian pulse problem, with Zero-mean reconstructionand full mass matrix method in an unstructured grid

5.2. Convected monopole

This case reproduces the example of [2]. The radiation of a monopolesource is computed in a subsonic mean flow, with Mach number Mx = 0.5.The source is located at xs = ys = 0, and is defined as:

Sp =1

2exp

(−ln(2)

(x− xs)2 + (y − ys)

2

2

)sin (ωt)× [1, 0, 0, 1]T (32)

where the angular frequency is ω = 2π/30 and t is the time coordinate.The wave length is λ = 30 units, and the computational domain is a squarewith 200 units for each side. The source term is made dimensionless with[ρ0c0/∆x, 0, 0, ρ0c

30/∆x]

T. With the aim of testing the stability and the be-

haviour of the proposed method for the boundary conditions, an unstructuredgrid absorbing layer has been added. The absorbing layer is placed from theboundary of the computational domain to x = ±300 and y = ±300. In

31

figure 7 (top) it is shown the grid used for the resolution of this problem. Tobuild this grid, 800 equally spaced nodes at the circumference of the computa-tional domain are used and 120 nodes at the outer boundaries circumference.

In addition to the absorbing boundary condition given by equation (28),the shape filter parameters of the absorbing layer are κx = κy = 8 [24].A 5th order mass matrix based FV-MLS solver is used for this example.Two acoustic waves propagate upstream and downstream of the source, anddue to the effect of the mean flow, the apparent wavelength is modified and itis different upstream (λ1 = (1−Mx) λ) and downstream (λ2 = (1 +Mx) λ)of the source.

In figure 8 pressure isocontours for different non-dimensional times t areshown. The pressure profile along axis y = 0 at time t = 270 is reproducedin figure 9, and also match the results in [2].

32

Figure 7: Convected monopole - M=0.5, 200× 200 grid, t = 270 at y = 033

a) b) c)

d) e)

Figure 8: Convected monopole - M=0.5, 200 × 200 grid. a) t=60, b) t=90, c) t=150, d)t=210, e) t=270

34

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

-100 -80 -60 -40 -20 0 20 40 60 80 100

p

x

Exact

with mass-matrix, 5th order

Figure 9: Convected monopole - M=0.5, 200× 200

In order to check the stability of the boundary conditions, we let thecomputations to continue until 180 periods of the source. This time corre-sponds to the travel of the wave until the four outer boundaries. Comparingthe pressure field with the one corresponding to t = 270 (9 source periods),it is observed that there is no change in the solution. The acoustic waveis completely dissipated by the buffer zone when it lefts the computationaldomain, see figure (7) (top).

5.3. Two cylinders scattering problem

This case can be considered as one of the most difficult validation prob-lems in CAA. It concerns the scattering of sound generated by a time depen-dent acoustic source from two circular cylinders. This problem is presentedin [55] as providing a stringent test of the ability of high order CAA codes tohandle complex geometries. It is presented by [55] also as a demonstrationof numerical robustness, long-time stability and suitability of far-field treat-ments in presence of multiple scattering bodies (two in our case).

35

The source is located at xs = ys = 0, and is defined as:

Sp = exp

(−ln(2)

(x− xs)2 + (y − ys)

2

b2

)× sin (ωt)× [0, 0, 0, 1]T (33)

where the angular frequency is ω = 8π, b = 0.2 and t is the time coordinate.Which corresponds to a period T = 0.25 and a wave length λ = 0.25. Thesource term is made dimensionless with [ρ0c0/∆x, 0, 0, ρ0c

30/∆x]

T.

The global computational domain, including the buffer zone (2 unitsthickness), is a rectangle with dimensions 22 × 12, the two cylinders havediameters: Dcylinder1 = 1 and Dcylinder2 = 0.5, they are located respectivelyat (−4, 0) and (4, 0) from the source, see figure below.

×S

4 4

Dcylinder1 = 1Dcylinder2 = 0.5

Figure 10: Two cylinder scattering problem

To avoid the spurious reflections caused by the large wave front emittedduring the first periods, the source is ramped up over the first 4 periods usinga cubic function φ(t) given by:

φ(t) = min

(1,

(t

4T

)3)

(34)

The grid is made such as we obtain almost 10 points per wave length usingtriangular cells. A constant time step of 0.001 is used with 40000 steps corre-sponding to 160 cycles. The 5th order mass matrix based FV-MLS solver isused for this example. The RMS pressure, computed in the last 16 periods,on the two cylinders surfaces and along the center line (y=0) is compared to

36

the analytical solutions in figures (11) and (12).

Figure 11: RMS pressure along the both cylinders

With the current grid resolution, figures (11) and (12) show, in general, agood agreement between computations and the exact solution. Oscillationsare observed for RMS pressure values smaller than 2×10−6 in figure (11). Infigure (12), we observe also the same phenomenon. The appearance of suchoscillations has been reported by many authors [55]. The grid resolutionmay be the cause of these oscillations. In fact, in [56] it is shown that theoscillations disappear with grid refinement.

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RMS pressure

8.315e-07

6.144e-06

4.54e-05

0.0003355

0.002479

0.01832

x-10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 12: RMS pressure along y = 0

5.4. Flow past a circular cylinder at ReD = 3900

In this section we show the application of the method presented in this pa-per to a stiff flow problem represented by unsteady Navier-Stokes equations.The considered case concerns a 2D flow over a cylinder at Mach 0.2 with aReynolds number of 3900. The present computational results are comparedto other numerical studies and also to experiments in order to assess theaccuracy of our approach. To solve this case, we used the 3rd order implicitfull-mass matrix FV-MLS method presented before. The 3rd order of spaceaccuracy was chosen according to the results of [54].

A value of the kernel parameter κ = 3 is chosen. We have used anunstructured triangular grid, with 200 elements on the surface of the cylinderand 214 points on the external circular boundary with a growth rate of 1.05.The final number of elements is 44042. A close view of the grid near thesurface of the cylinder is shown in figure (13).

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Figure 13: Computational domain unstructured grid

Results for the drag coefficient (Cd), the back pressure coefficient (Cpb),the separation angle θ and the Strouhal (St) number and are shown in table5.4. Figure (14) shows the vortex field. The obtained vortex shedding istypical to 2D simulations and it is observed only using LES approach. In fact,URANS fails to capture this feature [57]. The simulation with the proposedmethodology obtains accurate results for a 2D simulation and compares wellthe LES simulations with few elements in the z direction. The over predictionof the computed coefficients and of the Strouhal numbers is due to the lackof three-dimensionality in the computations.

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Data from Cd −Cpb θ StFV-MLS 3rd order - 10 cycles 1.280 0.960 87.8 0.25FV-MLS 3rd order - 30 cycles 1.268 0.898 87.4 0.25

2D URANS [58] 1.59 1.96 - 0.242D (Hybrid LES-RANS) [57] 1.5 - 86.5 0.25

3D LES (nz = 4) [58] 1.55 1.86 - 0.22Experiments∗ 0.99± 0.05 0.88± 0.05 86± 2 0.215± 0.005

Table 4: Results for the simulation of the turbulent flow past a cylinder.∗ The experimentalvalues have been taken from [59] and references therein.

Figure 14: Instantaneous vorticity field at Re=3900

6. Conclusion

A higher-order FV-MLS method was successfully used to solve unsteadyflow problems on unstructured grids. Two high-order preserving approacheswere tested, using independently an explicit time integration scheme for zero-mean reconstruction method and an implicit time integration scheme formass-matrix formulation.

RK46-NL was used for the explicit scheme and a BDF preconditionedGMRES solver was used for the implicit scheme. An order of accuracy study

40

was performed for both approaches. The study reveals that zero-mean re-construction approach was limited to the 3rd order while the mass-matrixformulation did not present this limitation. We proved that the later ap-proach can reach at least the 5th order of accuracy and nothing seems tolimit it, except, of course, the computational resources. For orders 2 and 3,zero-mean reconstruction method obtains more accurate results due to thewell optimized time integration scheme.Some test cases were analysed for the resolution of the Linearized EulerEquations (LEEs) with uniform mean flow and Navier-Stokes equations. Forthe convected monopole and the two cylinder scattering problems, only 5thorder mass-matrix based FV-MLS was used. The obtained results were ingood agreement with analytical solution. The two cylinder scattering casepresented some numerical oscillation for RMS pressure values smaller than2 × 10−6 due probably to the grid size. Refining the grid should overcomethis problem. In the last case, the Navier-Stokes equations were solved tosimulate flow past a circular cylinder at ReD = 3900 and Mach 0.2. Theobtained results were in good agreement with other 2D simulations and ex-periments. This is a stiff problem that shows the viability of this approachto solve this kind of flows.

After this study it is clear that the best kind of problems to apply themethodology we propose are stiff flow problems, where explicit time-steppingis not competitive. In those cases (stiff problems), the mass matrix does notintroduce any additional computational cost, and thus the advantages of theproposed higher order scheme is clear. Moreover, the proposed method donot need any additional degrees of freedom. Note that the present approachallows us to use of all the well-developed Riemann solvers technology. An-other advantage is that it is easy to implement to improve any existing finitevolume code. On the other hand, the present methodology is slightly lesscompact than other existing approaches, as CDG scheme for example. More-over, the direct discretization of high order viscous fluxes is a clear advantageof the FV-MLS methodology over other existing higher order methods.

In this paper, only implicit time solver was tested with the mass-matrixbased FV-MLS. Like most implicit solvers, time consuming convergence isthe major weakness. The use of a mass matrix formulation is justified fornon-stiff problems (LEE), since it allows us to achieving very high order ofaccuracy on unstructured grids. The explicit zero-mean method is blocked

41

at 3rd order.However, to take advantage from the low dissipation and low dispersion ex-plicit schemes, we need to inverse the mass matrix or use lumping techniques.The second solution seems more attractive but one needs to construct anadapted procedure for MLS formulation. This is currently in progress.

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