Improved High-Order Spe tral Finite Volume Method

Implementation for Aerodynami Flows

Carlos Breviglieri ∗ and Maximiliano A. F. Souza ∗

Instituto Te nológi o de Aeronáuti a, São José dos Campos 12228-900, Brazil

Edson Basso† and João Luiz F. Azevedo ‡

Instituto de Aeronáuti a e Espaço, São José dos Campos 12228-903, Brazil

The present work has the obje tive of demonstrating improvements in performan e and apabilities of a spe tral nite volume s heme implemented in a ell entered nite volume ontext for unstru tured meshes. The 2-D Euler equations are onsidered to represent theows of interest. The spatial dis retization s heme is developed to a hieve high resolutionfor ow problems governed by hyperboli onservation laws. Roe's ux dieren e splittingmethod is used as the numeri al approximate Riemann solver. Several appli ations areperformed in order to assess the method apability ompared to data available in theliterature and also ompared to an weighted essentially non-os illatory (WENO) s heme.There is good agreement with the omparison data and e ien y improvements over theWENO method are observed. The proposed improvements for the method in lude animpli it time mar h algorithm, exa t high-order boundary representation and an e ienthierar hi al moment limiter to treat ow solution dis ontinuities.

I. Introdu tion

Over the past several years, the Computational Aerodynami s Laboratory of Instituto de Aeronáuti a

e Espaço (IAE) has been developing CFD solvers for two and three dimensional systems.1,2 One resear h

area of the development eort is aimed at the implementation of high-order methods suitable for problems

of interest to the Institute, i.e., external high-speed aerodynami s. Some upwind s hemes su h as the van

Leer ux ve tor splitting s heme,3 the Liou AUSM+ ux ve tor splitting s heme4 and the Roe ux dieren e

splitting s heme5 were implemented and tested for se ond-order a ura y with a MUSCL re onstru tion.6

However, the nominally se ond-order s hemes presented results with an order of a ura y smaller than

expe ted in the solutions for unstru tured grids. Aside from this fa t, it is well known that total variation

diminishing (TVD) s hemes have their order of a ura y redu ed to rst order in the presen e of sho ks due

to the ee t of limiters.

This observation has motivated the group to study and to implement essentially non-os illatory (ENO)

and weighted essentially non-os illatory (WENO) s hemes in the past.7 However, as the intrinsi re on-

stru tion model of these s hemes relies on gathering neighbouring ells for polynomial re onstru tions for

ea h ell at ea h time step, both s hemes were found to be very demanding on omputer resour es for res-

olution orders greater than three, in 2-D, or anything greater than 2nd order, in 3-D. This fa t motivated

the onsideration of the spe tral nite volume method, as proposed by Wang and o-workers,813 as a more

e ient alternative. Su h method is expe ted to perform better than ENO and WENO s hemes, ompared

to the overall ost of the simulation, sin e it diers on the re onstru tion model applied and it is urrently

extended up to 4th-order a ura y in the present work. The spe tral nite volume (SFV) method is already

in use by the authors and numeri al results have been previously published.14 Although the expe t order of

∗Graduate Student, Department of Computer & Ele troni Engineering, Comando-Geral de Te nologia Aeroespa ial,CTA/ITA/IEC, Brazil.

†Senior Resear h Engineer, Comando-Geral de Te nologia Aeroespa ial, CTA/IAE/ALA, Brazil. E-mail: bassoiae. ta.br.‡Senior Resear h Engineer, Comando-Geral de Te nologia Aeroespa ial, CTA/IAE/ALA, Brazil. Asso iate Fellow AIAA.

E-mail: azevedoiae. ta.br.

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27th AIAA Applied Aerodynamics Conference22 - 25 June 2009, San Antonio, Texas

AIAA 2009-4119

Copyright © 2009 by C. Breviglieri, M.A.F. Souza, E. Basso, J.L.F. Azevedo. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

a ura y is obtained for the 2nd, 3rd and 4th order SFV methods, a signi ant deterioration in onvergen e

rate for the fourth order simulations was observed in Ref. 14, espe ially in the presen e of sho k waves.

Hen e, it is expe ted that su h behaviour an be over ome by the use of an impli it time mar h algorithm.15

Furthermore, the type of mesh element urrently adopted does not support quadrati or ubi boundary

representations. Su h drawba k typi ally translates into ine ien y, for ing the mesh to have mu h more

elements simply to represent the geometry.

Another important aspe t of a ompressible ow solver is the limiter te hnique employed. The use of

limiters is ne essary when the ow solution ontains dis ontinuities in order to remove spurious os illations

that may eventually lead to divergen e of the numeri al solution. Previous work on limiter implementations

for high-order methods14 was based on problem-dependent parameters to nd out whi h elements need

limiting, whi h an limit too many or too few elements on the solution. In the rst ase, the order of the

method is seriously redu ed and, in the se ond one, divergen e an o ur. To ir umvent this drawba k,

the present work uses a parameter-free generalized moment limiter16 to deal with dis ontinuities, whi h, as

suggested, does not require input onstants from the user. The present resear h is the rst work, on the

authors' knowledge, to apply this limiter te hnique for the SFV method. The numeri al solver is urrently

implemented for the solution of the 2-D Euler equations in a ell entered nite volume ontext for triangular

meshes, with an impli it LU-SGS s heme for time integration.

The remainder of the paper is organized as follows. In Se tion II, the theoreti al formulation is detailed,

in luding spatial and time integration methods. In Se tion III, the general SFV method re onstru tion pro-

ess is presented. Details are given for linear, quadrati and ubi polynomial re onstru tions for triangular

mesh elements. Se tion IV presents the formulation for high-order boundary representation. Moreover, a

new te hnique employing exa t geometry representation is proposed, whi h seems to be another ontribution

from the work. In Se tion V, the limiter formulation is dis ussed in detail. The limiter is of great importan e

for the problems of interest to the authors. In Se tion VI, numeri al results are presented. Con lusions and

remarks for future work are given in Se tion VII.

II. Theoreti al Formulation

A. Governing Equations

In the present work, the 2-D Euler equations are solved in their integral form as

∂

∂t

∫

V

QdV +

∫

V

(∇ · ~P )dV = 0 , (1)

where ~P = Eı + F . The appli ation of the divergen e theorem to Eq. (1) yields

∂

∂t

∫

V

QdV +

∫

S

(~P · ~n)dS = 0 . (2)

The ve tor of onserved variables, Q, and the onve tive ux ve tors, E and V , are given by

Q =

ρ

ρu

ρv

et

, E =

ρu

ρu2 + p

ρuv

(et + p)u

, F =

ρv

ρuv

ρv2 + p

(et + p)v

. (3)

The standard CFD nomen lature is being used here. Hen e, ρ is the density, u and v are the Cartesian

velo ity omponents in the x and y dire tions, respe tively, p is the pressure, and et is the total energy per

unit volume. The system is losed by the equation of state for a perfe t gas

p = (γ − 1)

[

ei −1

2ρ(u2 + v2)

]

, (4)

where ei is the spe i internal energy, and the ratio of spe i heats, γ, is set as 1.4 for all omputations in

this work. In the nite volume ontext, for xed meshes, Eq. (2) an be rewritten for the i-th mesh element

as

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∂Qi

∂t= − 1

Vi

∫

SVi

(~P · ~n)dS , (5)

where Qi is the ell averaged value of Q at time t and Vi is the volume, or area in 2-D, of the i-th mesh

element.

B. Spatial Dis retization

The spatial dis retization pro ess determines a k-th order dis rete approximation to the integral in the right-

hand side of Eq. (5). In order to solve it numeri ally, the omputational domain, Ω, with proper initial and

boundary onditions, is dis retized into N non-overlapping triangles, the spe tral volumes (SVs), su h that

Ω =

N⋃

i=1

SVi. (6)

One should observe that the spe tral volumes ould be omposed by any type of polygon, given that it is

possible to de ompose its bounding edges into a nite number of line segments ΓK , su h that

SVi =⋃

ΓK . (7)

In the present paper, however, the authors assume that the omputational mesh is always omposed of

triangular elements. Hen e, although the theoreti al formulation is presented for the general ase, the a tual

SV partition s hemes are only implemented for triangular grids.

The boundary integral in Eq. (5) an be further dis retized into the onve tive operator form

C(Qi) ≡∫

SVi

(~P · ~n)dS =

nf∑

r=1

∫

Ar

(~P · ~n)dS, (8)

where nf is the number of fa es, or edges in 2-D, of SVi, and Ar represents the area, or the length in 2-D,

of the r-th edge of the SV. Given the fa t that ~n is onstant for ea h line segment, the integration on the

right side of Eq. (8) an be performed numeri ally with a k-th order a urate Gaussian quadrature formula

∫

SVi

(~P · ~n)dS =

nf∑

r=1

nq∑

q=1

wrq~P (Q(xrq, yrq)) · ~nrAr + O(Arh

k) , (9)

where (xrq, yrq) and wrq are, respe tively, the Gaussian quadrature points and the weights on the r-th edge

of SVi, nq = integer((k + 1)/2) is the number of quadrature points required on the r-th edge, and h will be

dened in the forth oming dis ussion. For the se ond-order s hemes, one Gaussian quadrature point is used

in the integration. Given the oordinates of the end points of the element edge, z1 and z2, one an obtain

the quadrature point as the middle point of the segment onne ting the two end points, G1 = 12 (z1 + z2).

For the third and fourth order s hemes, two Gaussian points are ne essary along ea h line segment. Their

values are given by

G1 =

√3 + 1

2√

3z1 +

(

1 −√

3 + 1

2√

3

)

z2 and G2 =

√3 + 1

2√

3z2 +

(

1 −√

3 + 1

2√

3

)

z1., (10)

Using the method des ribed above, one an ompute values of Qi for instant t for ea h SV. Due to the

dis ontinuity of the re onstru ted values of the onserved variables over SV boundaries, one must use a

numeri al ux fun tion to approximate the ux values along the ell boundaries.

The previously des ribed pro edure follows exa tly the standard nite volume method. For a given order

of spatial a ura y, k, for Eq. (8), using the SFV method, ea h SVi element must have at least

m =k(k + 1)

2(11)

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degrees of freedom (DOFs). This orresponds to the number of ontrol volumes (CVs) that SVi shall be

partitioned into. If one denotes by CVi,j the j-th ontrol volume of SVi, the ell-averaged onservative

variables, q, at time t, for CVi,j is omputed as

qi,j =1

Vi,j

∫

CVi,j

q(x, y)dV , (12)

where Vi,j is the volume of CVi,j . On e the ell-averaged onservative variables, or DOFs, are available for

all CVs within SVi, a polynomial, pi(x, y) ∈ P k−1, with degree k − 1, an be re onstru ted to approximate

the q(x, y) properties inside SVi, i.e.,

pi(x, y) = q(x, y) + O(hk−1), (x, y) ∈ SVi, (13)

where h represents the maximum edge length of all CVs within SVi. The polynomial re onstru tion pro ess

is dis ussed in details in the following se tion. For now, it is su ient to say that this high-order re on-

stru tion is used to update the ell-averaged state variables at the next time step for all the CVs within

the omputational domain. Note that this polynomial approximation is valid within SVi and the use of

numeri al uxes are ne essary a ross SV boundaries.

Integrating Eq. (5) in CVi,j , one an obtain the integral form for the CV mean state variable

dqi,j

dt+

1

Vi,j

nf∑

r=1

∫

Ar

(~f · ~n)dS = 0, (14)

where ~f represents the E and F uxes, and nf is the number of edges of CVi,j . The numeri al integration

an be performed with a k-th order a urate Gaussian quadrature formulation, similarly to the one used for

the SV elements in Eq. (9).

As previously stated, the ux integration a ross SV boundaries involves two dis ontinuous states, to the

left and to the right of the edge. This ux omputation an be arried out using an exa t or approximate

Riemann solver, or even a ux splitting pro edure, whi h an be written in the form

~f (q(xrq, yrq)) · ~nr ≈ fRiemann (qL(xrq, yrq), qR(xrq, yrq), ~nr) , (15)

where qL is the onservative variable ve tor obtained by the pi polynomial applied at the (xrq, yrq) oordinatesand qR is the same ve tor obtained with the pnb polynomial in the same oordinates of the edge. Note that

the nb subs ript represents the element to the right of the edge, whereas the i subs ript denotes the CV to

its left. As the numeri al ux integration in the present paper is based on one of the forms of a Riemann

solver, this is the me hanism whi h introdu es the upwind and arti ial dissipation ee ts into the method,

making it stable and a urate. In this work, the authors have used the Roe ux dieren e splitting method5

to ompute the numeri al ux, i.e.,

fRiemann = froe(qL, qR, ~n) =1

2

[

~f(qL) + ~f(qR) −∣

∣B∣

∣ (qR − qL)]

. (16)

Here,∣

∣B∣

∣ is the Roe dissipation matrix in the dire tion normal to the edge. The B matrix, in the edge-normal

dire tion, is dened as

B = nx

∂E

∂q+ ny

∂F

∂q. (17)

The B matrix has four real eigenvalues, namely, λ1 = λ2 = vn, λ3 = vn + a, λ4 = vn − a, where vn is the

velo ity omponent normal to the edge and a is the speed of sound. Let T be the matrix omposed of the

right eigenve tors of B. Then, this matrix an be diagonalized as

T−1BT = Λ , (18)

where Λ is the diagonal matrix omposed of the eigenvalues of B, whi h an be written as

Λ = diag [vn, vn, vn + a, vn − a] . (19)

∣

∣B∣

∣ matrix is formed as∣

∣B∣

∣ = T∣

∣Λ∣

∣T−1, (20)

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where Λ and T are al ulated as a fun tion of the Roe averaged properties.5 Furthermore,∣

∣Λ∣

∣ uses the

magnitude of the eigenvalues.

Finally, one ends up with the semi-dis rete SFV s heme for updating the CVs, whi h an be written as

dqi,j

dt= − 1

Vi,j

nf∑

r=1

nq∑

q=1

wrqfRiemann(qL(xrq, yrq), qR(xrq, yrq), ~nr)Ar , (21)

where the right hand side of Eq. (21) is the equivalent onve tive operator, C(qi,j), for the j-th ontrol

volume of SVi. It is important to emphasize that some edges of the CVs, resulting from the partition of the

SVs, lie inside the SV element in the region where the polynomial is ontinuous. For su h edges, there is no

need to ompute the numeri al ux, as des ribed above. Instead, one uses analyti al formulas for the ux

omputation, i.e., without numeri al dissipation.

C. Temporal Dis retization

The onvergen e behaviour of high-order methods, su h as the SFV method, is generally poor with expli it

time mar hing approa hes. In order to obtain the steady state solution of the ow from an initial ondition,

a relaxation s heme is ne essary. The approa h typi ally used in the present resear h group has been

to resort to expli it, multi-stage, Runge-Kutta time-stepping methods. The main advantages of su h an

approa h are that it is easy to implement and the memory requirements are quite modest. Hen e, the

urrent produ tion version of the ode uses a 3-stage TVD Runge-Kutta s heme for time integration.7

However, adequate solution onvergen e hara teristi s, espe ially for the higher-order implementations,

di tate that an impli it time integrator should be implemented. Therefore, an impli it LU-SGS s heme is

implemented in the ontext of the present work.

Equation (14) an be re ast in the semi-dis rete form as

Vi,j

∂qi,j

∂t= −Ri,j (22)

where Ri,j is the right-hand side residual for the j-th CV of the i-th SV and it tends to zero as the simulation

onverges to a steady-state solution. Using the impli it Euler method for time integration, Eq. (22) an be

written in dis rete form as

Vi,j

∆qni,j

∆t= −Rn+1

i,j (23)

where ∆t is the time in rement and ∆qn = qn+1 − qn. The above equation an be linearized in time as

Vi,j

∆qni,j

∆t= −Rn

i,j −∂Rn

i,j

∂q∆qn

i,j . (24)

The term ∂R/∂q represents the Ja obian matrix. Writing the equation for all ontrol volumes, one obtains

the delta form of the ba kward Euler s heme

A∆q = −R (25)

where

A =V

∆tI+

∂Rn

∂q(26)

where I is the identity matrix.

In order to redu e the number of non-zero entries in the Ja obian matrix and to simplify the linearization

pro ess, only a rst-order representation of the numeri al uxes is linearized. This results in a graph for the

sparse matrix whi h is identi al to the graph of the unstru tured mesh. Hen e, the Ja obian matrix entries

an be omputed and stored over a loop on the mesh edges. Therefore, the following simplied ux fun tion

is used to obtain the left-hand side Ja obian matrix in Eq. (26),

Ri(qi, qnb, ~nr) =1

2

[

~f(qi, ~nr) + ~f(qnb, ~nr) − |λ|(qnb − qi)]

(27)

for the r-th edge that shares the i and nb ontrol volumes. A s alar dissipation model is used, where

|λ| = |vn| + a . (28)

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One should note that the dissipation on the ux fun tion is approximated by the Ja obian matrix spe tral

radius. The linearization of Eq. (27) yields

∂Ri

∂qi

=1

2(J(qi) + |λ|I)

∂Ri

∂qnb

=1

2(J(qnb) − |λ|I)

(29)

where J = ∂f/∂q is the Ja obian of the invis id ux ve tors in the dire tion normal to the edge.

Using an edge-based data stru ture the Ja obian matrix is stored in lower, upper and diagonal ompo-

nents, whi h are omputed as

L =1

2[−J(qi, ~nr) − |λ|I]

U =1

2[J(qnb, ~nr) − |λ|I]

D =V

∆tI+

∑

nb

1

2[J(qi, ~nr) + |λ|I] .

(30)

Note that L,U and D represent the stri t lower, upper and diagonal matri es, respe tively. Equation (25)

represents a system of linear simultaneous algebrai equations that needs to be solved at ea h time step.

The iterative LU-SGS solution method is employed, along with a mesh renumbering algorithm,17 and the

system is solved in two steps, a forward and ba kward sweep,

(D + L)∆q∗ = R

(D + U)∆q = D∆q∗.(31)

III. Spe tral Finite Volume Re onstru tion

A. General Formulation

The evaluation of the onserved variables at the quadrature points is ne essary in order to perform the ux

integration over the mesh element fa es. These evaluations an be a hieved by re onstru ting onserved

variables in terms of some base fun tions using the DOFs within a SV. The present work has arried out

su h re onstru tions using polynomial base fun tions, although one an hoose any linearly independent set

of fun tions. Let Pm denote the spa e of m-th degree polynomials in two dimensions. Then, the minimum

dimension of the approximation spa e that allows Pm to be omplete is

Nm =(m + 1)(m + 2)

2. (32)

In order to re onstru t q in Pm, it is ne essary to partition the SV into Nm non-overlapping CVs, su h that

SVi =

Nm⋃

j=1

CVi,j . (33)

The re onstru tion problem, for a given ontinuous fun tion in SVi and a suitable partition, an be stated

as nding pm ∈ Pm su h that∫

CVi,j

pm(x, y)dS =

∫

CVi,j

q(x, y)dS. (34)

With a omplete polynomial basis, el(x, y) ∈ Pm, it is possible to satisfy Eq. (34). Hen e, pm an be expressed

as

pm =

Nm∑

l=1

blel(x, y), (35)

where e is the base fun tion ve tor, [e1, · · · , eNm], and b is the re onstru tion oe ient ve tor, [b1, · · · , bNm

]T .The substitution of Eq. (35) into Eq. (34) yields

1

Vi,j

Nm∑

l=1

bl

∫

CVi,j

el(x, y)dS = qi,j . (36)

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If q denotes the [qi,1, · · · , qi,Nm]T olumn ve tor, Eq. (36) an be rewritten in matrix form as

Sb = q, (37)

where the S re onstru tion matrix is given by

S =

1Vi,1

∫

CVi,1e1(x, y)dS · · · 1

Vi,1

∫

CVi,1eNm

(x, y)dS

... · · ·...

1Vi,Nm

∫

CVi,Nm

e1(x, y)dS · · · 1Vi,Nm

∫

CVi,Nm

eNm(x, y)dS

(38)

and, then, the re onstru tion oe ients b an be obtained as

b = S−1q, (39)

provided that S is non-singular. Substituting Eq. (39) into Eq. (34), pm is, then, expressed in terms of shape

fun tions L = [L1, · · · , LNm], dened as L = eS−1, su h that one ould write

pm =

Nm∑

j=1

Lj(x, y)qi,j = Lq. (40)

Equation (40) gives the value of the onserved state variable, q, at any point within the SV and its boundaries,

in luding the quadrature points, (xrq, yrq). The above equation an be interpreted as an interpolation of a

property at a point using a set of ell averaged values and the respe tive weights, whi h are set equal to the

orresponding ardinal base value evaluated at that point.

On e the polynomial base fun tions, el, are hosen, the L shape fun tions are uniquely dened by the

partition of the spe tral volume. The shape and partition of the SV an be arbitrary, as long as the Smatrix is non-singular. The major advantage of the SFV method is that the re onstru tion pro ess does not

need to be arried out for every mesh element SVi. On e the SV partition is dened, the same partition

an be applied to all mesh elements and it results in the same re onstru tion matrix. That is, the shape

fun tions at similar points over dierent SVs have the same values. One an ompute these oe ients as a

pre-pro essing step and they do not hange along the simulation. This single re onstru tion is arried out

only on e for a standard element, for instan e, an equilateral triangle, and it an be read by the numeri al

solver as input. This is a major dieren e when ompared to k-exa t methods, or ENO and WENO s hemes,

for whi h every mesh element has a dierent re onstru tion sten il at every time step. Clearly, the SFV

is more e ient in this step. Re ently, several partitions for both 2-D and 3-D SFV re onstru tions were

studied and rened.11,18 For the present work, the partition s hemes are presented in the following se tions.

Moreover, the polynomial base fun tions for the linear, quadrati and ubi re onstru tions are listed in

Table 1. For more details regarding partition quality and stability analysis the interested reader is referred

to Refs. 14 and 19.

Table 1. Polynomial base fun tions.

Re onstru tion Order e

linear [ 1 x y ]

quadrati [ 1 x y x2 xy y2 ]

ubi [ 1 x y x2 xy y2 x3 x2y xy2 y3 ]

B. Linear Re onstru tion

For the linear SFV method re onstru tion, m = 1, one needs to partition a SV into three sub-elements, as

in Eqs. (11) and (32) and use the base ve tor as dened in Table 1. The partition s heme is given for a

standard element and it is uniquely dened for this ase. The stru tured aspe t of the CVs within the SVs

is used to determine neighborhood information and generate the global onne tivity data onsidering a hash

table sear h algorithm.20

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The linear partition is presented in Fig. 1(a). It yields a total of 7 points, 9 edges (6 are external edges

and 3 are internal ones) and 9 quadrature points. The linear polynomial for the SFV method depends only

on the base fun tions and on the partition shape. The integrals of the re onstru tion matrix in Eq. (38) are

obtained analyti ally for the triangular mesh elements.21 The shape fun tions, in the sense of Eq. (40), are

al ulated and stored in memory for the quadrature points, (xrq, yrq) of the standard element. Su h shape

fun tions have the exa t same value for the orresponding quadratures points of any other SV of the mesh,

provided they all have the same partition. There is one quadrature point lo ated at the middle of the every

CV edge.

C. Quadrati Re onstru tion

For the quadrati re onstru tion, m = 2, one needs to partition a SV into six sub-elements and use the base

ve tor as dened in Table 1. The partition s heme is also given in this work for a right triangle. The nodes of

the partition are obtained in terms of bary entri oordinates of the SV element nodes in the same manner

as the linear partition. The stru tured aspe t of the CVs within the SVs is used to determine neighborhood

information and generate the onne tivity table. The ghost reation pro ess and edge-based data stru ture

is the same as for the linear re onstru tion ase. The partition used in this work19 is shown in Fig. 1(b).

It has a total of 13 points, 18 edges (9 external edges and 9 internal ones) and 36 quadrature points. The

shape fun tions, in the sense of Eq. (40), are obtained as in the linear partition. The reader should note

that, in this ase, the base fun tions have six terms that shall be integrated. Again, these terms are obtained

exa tly21 and kept in memory. In this ase, two quadrature points are required per CV edge for numeri al

ux integration.

D. Cubi Re onstru tion

For the ubi re onstru tion, m = 3, one needs to partition the SV into ten sub-elements and to use the

base ve tor as dened in Table 1. The ghost reation pro ess and edge-based data stru ture is the same as

for the linear and quadrati re onstru tion ases. As a matter of fa t, the same algorithm used to perform

these tasks an be applied to higher order re onstru tions. The partition used in this work is the improved

ubi partition,19 presented in Fig. 1( ), and it has a total of 21 points, 30 edges (12 external edges and 18

internal ones) and 60 quadrature points The shape fun tions, in the sense of Eq. (40), are obtained as in

the linear partition in a pre-pro essing step. As with the quadrati re onstru tion, ea h CV edge has two

quadrature points for numeri al ux integration.

a b c

Figure 1. Triangular spe tral volume partitions for a-linear, b-quadrati , and - ubi re onstru tions.

IV. High-Order Boundary Representation

An advantage of high-order methods is that fewer number of unknowns are required to a hieve a given

level of a ura y than with lower order methods. Therefore, the omputational grids used in high-order

simulations an be onsidered oarse when ompared to those in se ond-order simulations. In se ond-order

CFD simulations, all urved boundaries are represented with line-segments or planar fa ets, sin e su h repre-

sentation is ompatible with the linear interpolations used in se ond-order ow solvers. In order to minimize

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the solution errors produ ed by this " rude" approximation of urved boundaries, many elements may be re-

quired to simply preserve the geometry with a reasonable a ura y. If this rude geometri representation is

used in high-order simulations, unne essarily ne grids may be required near urved boundaries to represent

the boundary with high delity. Obviously, this pra ti e an waste signi ant omputational resour es. A

mu h more desirable approa h is to represent urved boundaries with higher order polynomials ompatible

with the order of the re onstru tion. For example, quadrati or ubi polynomials should be used to approx-

imate boundaries in third and fourth order SFV s hemes, whi h employ quadrati and ubi polynomials to

represent the solution variables.

Following the usual pra ti e in high-order nite element method,22 isoparametri SVs an be used to

map SVs with urved boundaries into standard SVs. Hen e one an assume that a one-to-one transformation

exists between a general SV in the physi al spa e, (x, y), and the standard triangle in the omputational

domain, (ξ, η), i.e.,

ξ = ξ(x, y) (41)

η = η(x, y)

The partition of the SV is performed in the standard triangle, and the partition in the physi al domain is the

result of the inverse transformation from the omputational domain ba k to the physi al domain. Therefore,

a ne essary ondition for a valid transformation is that the inverse transformation exists. The DOFs for the

general SV are the CV-averaged state variables in the physi al domain,

qi,j =

∫

CVi,jq(x, y)dxdy

∫

CVi,jdxdy

=

∫

Dq(ξ, η)|J |dξdη∫

D|J |dξdη

, (42)

where J is the Ja obian matrix of the transformation, i.e., J = ∂(x,y)∂(ξ,η) . The re onstru tion problem an

be stated as onstru ting a (k − 1)-degree polynomial, pi(ξ, η), in the omputational domain, given the

CV-averaged state-variables in all the CVs of a SV, su h that

∫

D

pi(ξ, η)|J |dξdη = qi,j

∫

D

|J |dξdη . (43)

Given a basis set and a valid transformation, the re onstru tion problem an be solved either analyti ally

or numeri ally to obtain the re onstru tion oe ients. However, dierent urved SVs have dierent re on-

stru tion oe ients. Therefore, it is ne essary to store these oe ients for SVs with urved boundaries.

Sin e the number of urved SVs is expe ted to be small, when ompared to the total number of SVs in any

simulation, the memory ost for storing these oe ients is also small.

Geometry information is ne essary in order to orre tly position the nodes of the high-order SV element.

The authors have hosen to use exa t geometry data, instead of approximate data, to represent all boundaries

as splines in the standard IGES23 format.

V. Limiter Formulation

For the Euler equations, it is ne essary to limit some re onstru ted properties at ux integration points

in order to maintain stability and onvergen e of the simulation, if the ow solution ontains dis ontinuities.

The present limiter te hnique involves two stages. First, the solver must nd out and mark "troubled

ells" whi h are, in the se ond stage, limited. For the dete tion and limiting pro ess, the limiter employs a

Taylor series expansion for the re onstru tion24 with regard to the ell-averaged derivatives. The troubled

ells are, then, limited in a hierar hi al manner, i.e., from the highest-order derivative to the lowest-order

one. If the highest derivative is not limited, the original polynomial is preserved and so is the order of

the method at the element level. This limiter te hnique is apable of suppressing spurious os illations

near solution dis ontinuities without loss of a ura y at lo al extrema in smooth regions. Originally, this

limiter methodology was developed for the Spe tral Dieren e method in Ref. 16. In the present work, the

formulation is extended for the SFV method.

Several markers (or sensors) were developed and employed for unstru tured meshes over the past de ades.

For an in-depth review, the interested reader is referred to Ref. 25. The limiter marker used in the present

work is termed A ura y-Preserving TVD (AP-TVD) marker. One important aspe t is that the troubled- ell

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and limited properties are inherent to the SV element, and not to the CVs in whi h the ux al ulations are

performed. On e the SV element is onrmed as a troubled ell, its polynomial, based on CV ell-averaged

variables, an no longer be used in any ux integration point, be ause the property fun tion is no longer

smooth within su h SV. Hen e, it is of utmost importan e to limit as few SVs as possible. To that end, the

marker is designed to rst he k for the ux integration points within ea h SV and mark those that do not

satisfy the monotoni ity riterion. However, if an extremum is smooth, the rst derivative of the solution,

at su h point, should be lo ally monotoni . Hen e, on a se ond moment and using derivative information,

as des ribed in the forth oming dis ussion, the limiter sensor unmarks those SVs at lo al extrema that were

unne essarily marked as troubled ells. Therefore, for instan e, for a quadrati re onstru tion, the limiting

pro ess an be summarized in the following stages:

1. For a given spe tral volume, SVi, ompute the minimum and maximum ell averages using a lo al

sten il whi h in ludes its immediate fa e neighbors, i.e.,

Qmin,i = min

(

Qi , min16r6nf

Qr

)

Qmax,i = max

(

Qi , max16r6nf

Qr

) (44)

2. The i-th ell is onsidered as a possible troubled ell if

pi(xrq, yrq) > 1.001Qmax,i or pi(xrq, yrq) < 0.999Qmin,i . (45)

The 1.001 and 0.999 onstants are not problem dependent. They are simply used to over ome ma hine

error, when omparing two real numbers, and to avoid the trivial ase of when the solution is onstant

in the neighborhood of the spe tral volume onsidered. This step is performed in order to he k the

monotoni ity riterion.

3. Sin e the previous steps may ag more SVs than stri tly ne essary, the next operations attempt to

unmark SVs in smooth regions of the ow. Hen e, for a given marked spe tral volume, a minmod TVD

fun tion is applied to verify whether the ell-averaged se ond derivative is bounded by an estimate of

the se ond derivative using ell-averaged rst derivatives of the neighboring spe tral volumes. Su h

test is performed as:

• If the unit ve tor in the dire tion onne ting the entroids of the i-th and nb-th ells is denoted~l, where nb indi ates the fa e-neighbor of a marked SVi, the se ond derivative in su h dire tion

is dened as

Qll,i = Qxx,il2x + 2Qxy,ilxly + Qyy,il

2y ; (46)

• In a similar fashion, the rst derivatives in the same ~l dire tion, for both i-th and nb-th ells, an

be omputed as

Ql,i = Qx,ilx + Qy,ily ,

Ql,nb = Qx,nblx + Qy,nbly ;(47)

• Another estimate of the se ond derivative, in the ~l dire tion, an be obtained as

Qll,i =Ql,nb − Ql,i

|~ri − ~rnb|, (48)

where ~r is the entroid oordinate ve tor;

• A s alar limiter for this fa e is omputed a ording to

φ(2)i,nb = minmod

(

1,Qll,i

Qll,i

)

; (49)

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• The pro ess is repeated for the other fa es of SVi, and the s alar limiter for the SV is the minimum

among those omputed for the fa es, i.e.,

φ(2)i = min

nb

(

φ(2)i,nb

)

; (50)

• If φ(2)i = 1, the se ond derivatives are bounded, as previously dened, and, hen e, SVi is a tually

in a smooth region of the ow. Therefore, SVi is unmarked.

4. If the previous test is not satised, this means that the parti ular SVi spe tral volume should indeed

be limited. In this ase, the limiter for the rst derivative re onstru tion must also be omputed. The

al ulation pro edure follows the same approa h as for the se ond derivatives, and it an be summarized

as:

• An estimate of the rst derivative in the ~l dire tion is al ulated as

Ql,i =Qnb − Qi

|~ri − ~rnb|; (51)

• Su h estimate is ompared to the ell-average rst derivative in the~l dire tion, omputed a ording

to Eq. (47), in order to obtain the s alar limiter for the fa e as

φ(1)i,nb = minmod

(

1,Ql,i

Ql,i

)

; (52)

• As before, the s alar limiter for the ell is the minimum of those limiters omputed for the fa es,

i.e.,

φ(1)i = min

nb

(

φ(1)i,nb

)

. (53)

The ell-averaged derivatives for the i-th ell, ne essary to perform the above al ulations, are obtained

by solving a quadrati least-squares re onstru tion problem, for a 3rd-order s heme, or a ubi least-squares

re onstru tion problem, for a 4th-order s heme. Further details of the least-squares problem formulation an

be found in Ref. 26. Finally, the quadrati limited polynomial, whi h is used in order to obtain property

values at the quadrature points for a troubled SVi spe tral volume, is given by

plimitedi (xrq, yrq) = Qi + φ

(1)i

[

1

Vi

(QxMx + QyMy)i

]

+ φ(2)i

[

1

Vi

(

1

2QxxMx2 + QxyMxy +

1

2QyyMy2

)

i

]

.

(54)

The SV area moments an be omputed, up to the desired order of a ura y, by numeri al integration as

Mxmyn |i =

∫

SV

(xrq − xi)m(yrq − yi)

ndV . (55)

VI. Numeri al Results

The results presented here attempt to validate both the implementation of the data stru ture, temporal

integration, numeri al stability and resolution of the SFV method. The overall performan e of the method

is ompared with that of a WENO s heme implementation. For the results here reported, density is made

dimensionless with respe t to the freestream ondition and pressure is made dimensionless with respe t to

the freestream density times the freestream speed of sound squared. For the steady ase simulations, the

CFL number is set as a onstant value and the lo al time step is omputed using the lo al grid spa ing and

hara teristi speeds. For all test ases, the CFL number is set to 10+6.

All numeri al simulations were arried out on a dual- ore 3.0 GHz PC Intel64 ar hite ture, with Linux

OS. The ode is written in Fortran 95 language and the Intel Fortran ompiler R© with optimization agsa

is used. For the performan e omparisons whi h are presented in this se tion, all residuals are normalized

by the rst iteration residue. Moreover, the L2 norm is used in all residuals here reported, ex ept when

expli itly noted otherwise.

aCompiler ags: -O3 -assume buered_io -parallel

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p

1.81.71.61.51.41.31.21.110.90.8

Figure 2. Supersoni wedge ow unstru tured mesh with pressure ontours for the 3rd-order SFV method.

A. Wedge Flow

The rst test ase is the omputation of the supersoni ow eld past a wedge with half-angle θ = 15 deg,

based on data from Ref. 27. The omputational mesh has 2409 nodes and 4613 volumes and it is shown in

Fig. 2, along with numeri al pressure ontours obtained with the 3rd-order SFV method. For omparison

purposes, the 3rd-order SFV and WENO methods are used for this simulation. The leading edge of the

wedge is lo ated at oordinates (x, y) = (0, 0). The omputational domain is bounded along the bottom by

the symmetry plane of the wedge and the wedge surfa e. The inow boundary is lo ated at x = −0.5. Theoutow boundary is pla ed at x = 1.0. The fareld boundary is pla ed at y = 1.0, in order to guarantee

that it is lo ated above the oblique sho k. The analyti al solution gives the hange in properties a ross the

oblique sho k as a fun tion of the freestream Ma h number and sho k angle, whi h is obtained from the

θ − β −Mach relation. For this ase, a freestream Ma h number of M1 = 2.5 is used, and the oblique sho k

angle β is obtained as 36.94 deg. For the analyti al solution, the pressure ratio is p2/p1 ≈ 2.4675 and the

Ma h number past the sho k wave is M2 ≈ 1.8735. For the present simulations, the limiter is turned on, and

only elements in the sho k wave region are marked for limited re onstru tion, as one an observe in Fig. 3.

This gure shows (in red) the spe tral volumes in whi h the limiter is a tive for pressure re onstru tion in

the present test ase.

The numeri al solutions of the SFV method are in good agreement with the analyti al solution. The

third order results are ompared in Fig. 4, in terms of Ma h number distribution, with the analyti al values.

Note that the SFV s heme is the one that better approximates the jump in Ma h number at the leading edge.

The pressure ratio and Ma h number after the sho k wave for the third order SFV s heme are omputed as

p2/p1 ≈ 2.47 and M2 ≈ 1.88, performing some averaging of the omputational results right after the sho k in

the interior of the domain. It should be observed that the omputational results shown in Fig. 4 are slightly

dierent, be ause these are results at the wedge surfa e. The data in Fig. 4 indi ates that the SFV method

yields results whi h are slightly better than those al ulated by the WENO s heme, espe ially downstream

of the sho k wave. Moreover, the SFV method is mu h less expensive than the WENO al ulation, for the

same test ase, as indi ated in Figs. 5(a) and 5(b). It should be further emphasized that the third-order

SFV method a hieved a residual drop of twelve orders of magnitude, whereas the WENO method stalled

onvergen e after the rst few hundreds iterations, even with the use of an impli it time integration.

B. Ringleb Flow

The Ringleb ow simulation onsists in an internal subsoni ow, whi h has an analyti al solution for the

Euler equations derived with the hodograph transformation.28 The analyti al solution is used as initial

ondition for all simulations here dis ussed. The ow depends on the inverse of the stream fun tion, k, and

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Figure 3. Marked SVs for pressure limiting onsidering a 3rd-order SFV simulation of the supersoni wedgeow.

x

Mac

h

-0.5 -0.25 0 0.25 0.5 0.75 11.6

1.8

2

2.2

2.4

2.6

analyticalSFV 3rdWENO 3rd

Figure 4. Analyti al and numeri al supersoni wedge ow Ma h number distributions.

the velo ity magnitude, vt. In the present simulations, these parameters are hosen as k = 0.4 and k = 0.6,in order to dene the bounding walls, and vt = 0.35 to dene the inlet and outlet boundaries. For su h

onguration, the test ase represents an irrotational and isentropi subsoni ow around a symmetri blunt

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Iterations

log[

L2(

rhs(

1))

]

0 200 400 600 800 100010-13

10-11

10-9

10-7

10-5

10-3

10-1

101

SFV 3rdWENO 3rd

(a)

Time (sec)

log[

L2(

rhs(

1))

]

0 100 200 300 400 500 600

10-11

10-9

10-7

10-5

10-3

10-1

101

SFV 3rdWENO 3rd

(b)

Figure 5. Convergen e history for wedge ow with impli it 3rd-order SFV and WENO s hemes.

obsta le. In order to measure the order of the implemented SFV method, four meshes are onsidered for the

mesh renement study, orresponding to 128, 512, 2048 and 8192 spe tral volume elements. The analyti al

solution is omputed for all meshes in order to measure how lose the numeri al results are to the exa t

solution. The error with respe t to the analyti al solution is omputed using the density L1 and L∞ norms.

Figure 6 shows the 2048-element grid and the Ma h number ontours omputed in this grid with the fourth

order SFV method, using the orresponding high-order boundary representation.

It should be pointed out that the same numeri al test ase was studied in Ref. 14, onsidering only

the linear boundary representation. It was observed in that eort that su h boundary representation has

triggered the limiter for some of the simulations and, in some ases, it has even aused the al ulations to

diverge, espe ially for the ner meshes. For instan e, it was not possible to obtain a onverged solution

with the fourth order SFV simulations for su h ases. In the present work, however, whi h onsiders the

higher-order boundary representation, reasonable results are always obtained for this test ase, in luding

the simulations with the fourth order SFV method. As previously dis ussed, for the third order s heme,

a quadrati polynomial is used to represent the SV edges whi h lie along the geometry boundaries. In a

similar fashion, for the fourth order s heme, a ubi polynomial is employed instead, whi h is ompatible

with the internal polynomial order of ea h SV. Table 2 presents the L1 and L∞ density error norms for the

present al ulations with the high-order boundary representation. The table also shows the a tual measured

order of a ura y for the third and fourth order SFV methods. The orders of a ura y in the results shown

in Table 2 are al ulated as indi ated in Ref. 29. The a tual orders of a ura y here obtained are in good

agreement with those shown in the ited referen e.

C. RAE 2822 Airfoil

The transoni ow over a RAE 2822 airfoil with 2.31 deg. angle-of-atta k and freestream Ma h number

M∞ = 0.729 is also onsidered. In order to evaluate the present implementation against omputations

performed with a WENO s heme,7 two dierent simulations are performed for this test ase. Initially, a

simulation with a se ond-order WENO s heme, on a standard nite volume mesh whi h uses linear boundary

representation, is performed. The mesh for su h al ulation has 16, 383 nodes and 32, 399 elements, of whi h

340 elements lie on the airfoil surfa e, as ne essary to properly des ribe the airfoil geometry, espe ially on

the leading edge. This mesh is shown in Fig. 7. The se ond simulation is arried out for the third-order

SFV method with quadrati boundary representation. The approa h does not require a very ne mesh, as in

the rst test ase, be ause the geometry denition le is dire tly available to the solver. Moreover, only the

edges of SVs that are on the airfoil surfa e are mapped and stored for high-order boundary omputation. As

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Ameri an Institute of Aeronauti s and Astronauti s

mach

0.80.750.70.650.60.550.50.450.40.35

Figure 6. Ringleb ow mesh and Ma h number ontour results for fourth-order SFV method.

Table 2. A ura y assessment of SFV method for the Ringleb ow test ase.

Method Mesh elements L1 error L1 order L∞ error L∞ order

3rd order SFV 128 2.41E-02 - 2.14E-01 -

512 4.14E-03 2.54 2.45E-02 3.13

2048 6.27E-04 2.72 3.13E-03 2.97

8192 8.67E-05 2.85 3.60E-04 3.12

4th order SFV 128 5.77E-04 - 6.11E-03 -

512 6.48E-05 3.16 4.52E-04 3.76

2048 6.15E-06 3.39 5.68E-05 2.99

8192 6.87E-07 3.16 5.62E-06 3.33

dis ussed, only the boundary spe tral volumes with high-order urved edges need a spe ial treatment and

require the additional storage. Spe tral volumes in the interior of the domain are always treated through

the use of the standard partition s heme. The oarse mesh, used for this se ond test ase, has 2, 700 nodes

and 5, 265 elements, and it is shown in Fig. 8. This mesh has only 85 spe tral volume elements along the

airfoil surfa e. It should be noted, however, that, although the oarse mesh has mu h fewer elements, the

omputation is a tually arried over on 31, 590 CV elements for the third order SFV method. Therefore, the

two test ases are omparable.

The present test ases eviden e the large benet of using a high-order method for spatial dis retization.

The ability of obtaining an equivalent numeri al solution, with mu h less stringent mesh requirements than

would be ne essary to standard se ond order methods, is an important advantage. The numeri al Cp

results for the two test ases are presented in Fig. 9, together with orresponding experimental data.30 The

numeri al solution of the SFV method is in overall good agreement with the experimental data and with the

al ulations from the WENO s heme with the ner mesh, as shown in Fig. 9. Clearly, the experimental data

shows a smoothed transoni sho k, due to sho k waveboundary layer intera tion, whi h is an important

phenomenon for su h super riti al airfoils. The present omputations annot reprodu e su h physi s, sin e

the Euler equations are used. Furthermore, Fig. 10 presents the L2 norm of the density residuals as a fun tion

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Figure 7. Fine mesh used for se ond order WENO omputation.

Figure 8. Coarse mesh used for third order SFV omputation.

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of the number of iterations and of the a tual wall- lo k time. Su h results indi ate that the third-order SFV

method, with proper boundary treatment, outperforms the se ond-order simulation, espe ially in terms of

the number of iterations. It should be emphasized, however, that, although the residues in the WENO

al ulations do not drop to ma hine zero, the solution indeed onverges. It is lear that one would typi ally

feel more onfortable with residue histories that go to ma hine zero, as the present SFV method al ulations.

One should further observe that the omparison in terms of wall- lo k time indi ates that the omputational

osts for the two simulations are a tually omparable. In this parti ular ase, the added ost in the 3rd-

order SFV method al ulation omes about due to the need of limited re onstru tions around the sho k

wave and the additional operations asso iated with high-order boundary ux integrations. Nevertheless,

the omparison here onsiders a 2nd-order WENO and a 3rd-order SFV s hemes. If su h omparison had

onsidered 3rd-order s hemes in both ases, previous experien e7,14 indi ates that the SFV method would

outperform the WENO al ulation even in terms of wall- lo k time.

chord (x)

Cp

0 2 4 6 8 10

-1

-0.5

0

0.5

1

ExperimentalWENO 2ndSFV 3rd HO Boundary

Figure 9. Cp distributions for RAE 2822 airfoil at M∞ = 0.729 and α = 2.31 deg.

The limiter ability to mark only those SV elements stri tly along the dis ontinuity is another important

aspe t explored by the present test ase. The limited SVs in the nal iteration, for the solution with the 3rd-

order SFV method, are identied (in red) in Fig. 11. One an observe from this gure that the marked SVs

only orrespond to those in the numeri al sho k wave stru ture, as previously dis ussed in the present paper.

The pro ess of identifying SVs that need limiting and of a tually performing the re onstru tion of limited

properties an have a signi ant impa t on the omputational ost of the simulation. For some test ases,

su h pro edures an double the omputational time per iteration. For instan e, for every limited SV and

onsidering a third-order a ura y SFV s heme, there are 36 numeri al ux al ulations at quadrature points

that must be reevaluated with the limited properties and using the appropriate (approximate) Riemann

solver. Su h additional al ulation already represents almost twi e the omputational ost for a non-limited

SV, be ause only 18 numeri al ux evaluations are required per SV in the standard (non-limited) appli ation

of the 3rd-order SFV method. Therefore, by the present limiter implementation, only a few SV elements

are a tually limited during the omputation, rendering the method ompetitive. Moreover, the limiter

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Ameri an Institute of Aeronauti s and Astronauti s

Iterations

log[

L2(

rhs(

1))

]

0 2000 4000 6000 8000 10000

10-11

10-9

10-7

10-5

10-3

10-1

SFV 3rdWENO 2nd

(a)

Time (sec)

log[

L2(

rhs(

1))

]

0 100 200 300 400 500

10-12

10-10

10-8

10-6

10-4

10-2

100

SFV 3rdWENO 2nd

(b)

Figure 10. Convergen e histories for RAE 2822 airfoil with impli it 3rd-order SFV and 2nd-order WENOs hemes.

implementation here dis ussed is free from user-dependent input parameters, whi h is always a good feature

for a robust ow solver.

Figure 11. Limited SVs in the nal iteration for third order SFV simulation of the RAE 2822 airfoil ow.

VII. Con lusions

The appli ation of the high-order SFV method for aerodynami ow simulations is presented. An e ient

implementation, with regard to CPU time usage, is a hieved by employing several te hniques. One of su h

te hniques onsists in the use of an impli it time-mar h s heme, whi h allows for large time steps for the

steady-state al ulations here onsidered. The high-order boundary representation is also an important

aspe t, be ause it allows the use of a mu h oarser mesh, when ompared to the one required by se ond

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Ameri an Institute of Aeronauti s and Astronauti s

order s hemes to a hieve a similar level of geometry resolution. The present implementation of the limiter

te hnique, whi h extends, to SFV methods, ideas that have been tested previously on spe tral nite dieren e

s hemes, is an important ingredient of the method e ien y. The present limiter redu es the number of

limited spe tral volumes to a bare minimum, whi h redu es omputational osts and, at the same time, allows

for a more uniform high-order solution. Furthermore, a user-input-free limiter imp¨ementation ontributes

to enhan e the robustness of the ow solver.

The test ases addressed in the present work have indi ated that the present SFV method implementation

is able to a urately predi t the owelds of interest to the authors. In parti ular, the transoni airfoil

simulations are representative of the a tual ow features present in the relevant appli ations. In this test

ase, all three features previously indi ated have been brought to bear in the al ulations and they have

indeed yielded the expe ted behavior to the method. The te hniques added to the present implementation

have resolved all the problems the authors experien ed in the past with su h high-order SFV methods, as

dis ussed throughout the paper.

Further enhan ement of the methodology might ome from the use of matrix-free algorithms, in order

to relieve the memory requirements of the SFV method, parti ularly for the 4th-order s heme. It should be

observed that a 4th-order s heme requires a ubi polynomial re onstru tion and, hen e, ea h SV element

must be partitioned into ten CVs, rendering the a tual omputational mesh ten times larger than the original

SV grid. Clearly, a detailed study of the ee ts of the additional residue omputations per iteration, inher-

ently asso iated with matrix-free methods, must be performed in order to evaluate su h alternate approa h.

Moreover, in a ontinuous attempt to improve the approa h for implementing the limiter formulation, espe-

ially for the 4th-order SFV method, the authors have started to address the use of adaptive omputational

sten ils, whi h are typi al of ENO and WENO s hemes, in the onstru tion of the interpolation for SVs that

must be limited. Su h approa h has not been dis ussed in the present paper, be ause it still needs further

work in order to a hieve omparable e ien y as the formulation here presented. However, it is expe ted

that su h extension may lead to even more robust dis ontinuity apturing s hemes, and ir umvent the need

for the ostly al ulation of high-order ell-averaged derivatives.

VIII. A knowledgments

The authors gratefully a knowledge the partial support provided by Conselho Na ional de Desenvolvi-

mento Cientí o e Te nológi o, CNPq, under the Integrated Proje t Resear h Grant No. 312064/2006-3.

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