A DISCONTINUOUS hp FINITE ELEMENT METHODFOR THE SOLUTION OF THE EULER EQUATIONS

OF GAS DYNAMICS

Carlos Erik Baumann 1 and J. Tinsley Oden 2

Texas Institute for Computational and Applied MathematicsThe University of Texas at Austin

Austin, Texas 78712

Abstract

We present a discontinuous Galerkin technique for the solution of the Euler equa-tions of gas dynamics which produces a compact, higher-order-accurate, and stablesolver. The method involves a weak approximation of the conservation equations anda weak imposition of the Rankine-Hugoniot jump conditions across interelement anddomain boundaries. This discontinuous Galerkin approximation is conservative, andpermits the use of different polynomial order in each finite element, that can be adaptedaccording to the regularity of the solution. Moreover, the compadness of the formula-tion makes possible a consistent implementation of boundary conditions.

The numerical solutions of representative two- and three-dimensional problems sug-gest that the method is robust, capable of delivering high rates of convergence, andamenable to implementations in parallel computers.

1 Introduction

Most higher-order schemes for the solution of the Euler equations are applied to domains withsimple geometries. For those cases involving irregular meshes, particularly when adaptiverefinement is utilized, higher-order FD /FV schemes are often based on ad hoc argumentsthat may destroy the consistency or accuracy of tbe approximation [32][31][29][30][11][12].

Most FV schemes use piecewise constant representations of the field variables, and highaccuracy is obtained using reconstruction techniques [18][19][20][21][33][34] [13][14]. Thesetechniques are effective when the mesh has a good structure, but usually fail when the meshis irregular or the boundary conditions have a major influence on the solution.

The discontinuous Galerkin method (DGM) is a natural candidate for first-order partial dif-ferential equations. This method is relatively simple to code, requires only a data structureto describe the space discretization, and the representation of field variables is compact (ele-ment based). The method can deliver high accuracy without relying either on reconstruction

1Research Assistant2Director of TICAM, Cockrell Family Regents Chair in Engineering # 2

1

techniques (as in FV) or on large stencils (as in FD), which depend very strongly on thequality of the underlying mesh/grid.

The first study of discontinuous finite element methods for linear hyperbolic problems intwo dimensions was presented by Lesaint and Raviart in 1974 [23] (see also [22][24]) whoderived a pri01'i error estimates for special cases. Johnson and Pitkaranta [16] and Johnson[15] presented optimal error estimates using mesh dependent norms. Bey and Oden [6]extended the theory of discontinuous Galerkin methods by introducing hp-dependent normsand developed a priori and a posteriori error estimates for hyperbolic problems. Otherapplications can be found in [28][17][5][4].

Among the applications of the discontinuous Galerkin method to nonlinear systems of equa-tions, Cockburn el a/ [9][8][7]developed the TVB Runge-Kutta projection applied to generalconservation laws, Allmaras [1] solved the Euler equations using piecewise constant and piece-wise linear representations of the field variables. Lowrie [26] developed space-time discontin-uous Galerkin methods for nonlinear acoustic waves. Bey and Oden[3] presented solutionsto the Euler equations, and Lomtev, Quillen and Karniadakis[25] solved the Navier-Stokesequations discretizing the Euler fluxes with the DG method and using tl. mixed formulationfor the viscous fluxes.

This paper addresses many important issues which are fundamental for a successful appli-cation of the DGM to problems of industrial interest; among others, the selection of basisfunctions, the treatment of interelement discontinuities, and the selection of efficient time-marching schemes.

This paper is structured as follows: Section 2 introduces the problem and related notations,Section 3 presents the space discretization using the discontinuous Galerkin approximationin an infinite dimensional setting, and Section 4 in a finite dimensional setting. Finally,Section 5 presents a number of numerical experiments and Section 6 the conclusions.

2 Problem Statement

We shall now lay down the model problem and the associated notations in preparation fordeveloping and analyzing a weak formulation.

2.1 Governing equations

Let n be a bounded Lipschitz domain in IRd, sllch as the two-dimensional domain shown inFig. 1.

The governing equations for the conservation of mass, momentum, and energy, written invector form are

2

Figure 1: Notation for the domain and boundary.

au + div F = S,atU(x,O) = Uo(x),

.in n

at t = 0(1)

where U = (u 1, ... , Um) = U (x, t) E IRm is a vector of conservation variables, m = d + 2,Fi(U) = (fli, ...,fmi) E mm is the flux vector associated with the i-th space coordinate,and S is the source term.

To establish more specific notations of use later, we give the expressions of the governingequations in cartesian coordinates using (p, PUj, E), density, momentum, and total energy asthe conservation variables (summation on repeated indices is assumed):

(2)

where repeated indices i are summed throughout their range, E = p(1/2 UiUi + e) is thetotal energy of the fluid, e the internal energy, Ui the velocity associated with the i-th spacecoordinate, and 11 the static pressure.

3

It is well known that equations (1) hold only in sub domains of n x (0,00) where the spatialand temporal derivatives of V exist. There may exist surfaces within n x (0,00) where dis-continuities in V and in the fluxes occur, and on those surfaces the Rankine-Hugoniot jumpconditions take the place of the Euler equations as a statement of the conservation laws ..Following standard practice, we assume that n consists of sub domains where equations (1)hold in some weak sense separated by interfaces across which V and Fi exhibit jumps, andwe ultimately devise algorithms to locate approximately these surfaces supporting discon-tinuities. The various boundary conditions that apply to solutions of (1) can be found inmany publications, in particular, a description related to discontinuous approximations canbe found ill [2].

2.2 Properties of the flux vectors

The inviscid flux vectors Fi are homogeneous functions of degree one in the conservativevariables Vj therefore the fluxes can be written as Fi = Ai(V) V, where Ai(U) is theJacobian matrix.

Let F n(U) be the normal flux at any point on a boundary an with o.utwarc1 normal n, then

Fn = Fi ni i E [1, ... , d],

The flux vector F n(U) can be split into inflow and outflow components F~ and F;;, respec-tively, for example

where A is a diagonal matrix of eigenvalues of An, and the columns of the matrix R are thecorresponding eigenvectors. From a physical point of view, F~ and F;; represent the fluxesof mass, momentum, and energy leaving (+) and entering (-) the domain through an.Given that the approximation of field variables may be discontinuous across internal surfacesin n or across an, let us define

V± = lim U(x ± € n),e-+O+

where x is a point at a boundary, which can be real (e.g. bounding walls an) or artificia.l(e.g. interelement boundaries).

With this notation, F~(U-) is the flux in the direction n, and F;;(V+) is that in theopposite direction.

4

3 Discontinuous Galerkin approximation: space dis-cretization

Let P = {Ph(n)}h>O be a family of regular partitions of n c IRd into N == N(Ph) convexsubdomains ne, such that for 'Ph E P,

N(Phl

n = U ne, and ne n nf = 0 for e =1= f.e=l

The partitions Ph(n) are regular in the sense that if he = diam(ne), and

pe = sup { diam( s): sphere sene} ,

then for simplices

(3)

(4)

he < Cpe

(5)

and for quadrilaterals/hexahedras, if

fte = min {lengths of the edges of One} ,lie = min {interior angles between intersecting edges/faces of one} ,

then

(6)

Each sub domain ne has a Lipschitzian boundary ene, which consist of piecewise smootharcs plus vertices for d = 2, and piecewise smooth snrfaces and edges for d = 3.

For a partition Ph in this family, we introduce a broken space V (Ph) of admissible vectorsof conservation variables V = V (:v) as follows:

V (Ph) = {V : VV1oF(U2)EL1(ne);

VI' (F~e(V2) + F~JVt)) E L1 (One);V{VlI V2, U3} E V (Ph), V ne E Ph(O) }.

For a given sufficiently smooth initial condition data V0' and appropriate boundary con-ditions, the space discretization using the discontinuous Galerkin method can be stated as

5

follows:

Given Uo = Uo(:V), for t E (0, T), find

Here

U(., t) E V (Ph) X HI (0, T) such that U(:v,O) = Uo(:V), and

L {( l-VTaaU

dx + 1 wT (F~.(U-) + F~JU+)) dsn"E'Ph In. t an. (7)

and F~ are known in closed form for the usual flux vector and flux difference splittings (see[2] and the references therein).

Note that (7) renders a. conservative formulation. To show that (7) is globally conservative,let us pick a test function W = (Vi, ... ,vrn) such that

Vi(:V) = 1, i = l, ... ,m V:v E 0,

clearly W E V (Ph) . Substituting W in (7), we get

{ aaU dx + L ( (F~.(U-) + F~JU+)) ds = { S dx.Jn t n.E'Ph Jan" Jn

Noting that for any pair of adjoining elements (Oe, Of), the following identities hold:

and

(8)

we finally obtain

r aaU dx + r (F~(U-) + F;;(U+)) ds = { S dx,Jn t Jan Jn

which shows that the formulation is globally conservative.

To show that the formulation is also locally conservative, we select a generic weightingfunction

such that

6

i = 1, ... , m,

and substituting W in (7), we get

which represents the conservation equations at element level.

3.1 Equivalence of the weak formulation

In this section we invoke standard arguments to prove that any solution to problem (1) isalso a solution to the variational problem (7).

Assuming that the functions are sufficiently regular so that we may integrate by parts thelast term of the LHS of (7), we have

1 awT 1 T - 1 T aFi- -a Fj(U) dx == - W Fn,(U ) + W -a dx,n. Xj an. ne Xi

v De E Ph(n), and substituting the above equations in (7), we obtain

L 1 wT (aaU + aa

Fj- s) dx + L ( wT (F;;.(U+) - F;;.(U-)) ds == 0,

n ..e'Pll n. t Xl n.E'P1lJan,

V WE V (Ph)'

Clearly, any sufficiently smooth solution of the model problem (1) satisfies (7).

4 Discontinuous Galerkin approximation of the Eulerequations

For every element ne E Ph, the finite-dimensional space of real-valued shape functions istaken to be the space pp• (n) of polynomials of degree ~ Pe defined on the master element

n. Then we define

(9)

where Fn. is an invertible mapping from the master element n onto ne, see Fig. 2.

Let V p (Ph) be the following finite-dimensional space subspace of V (Ph) :

7

'I

"n l;c.

Figure 2: Mappings n ---+ !1e and discontinuous approximation.

and let the time domain be subject to a family of partitions:

A space-time discretization of the Euler equations ,,,,'itl1piecewise constant approximation intime can be stated as follows:

(10)

When the relevant time scales of the problem are close enough to justify the use of explicittime-marching schemes, this formulation is very convenient because the global mass matrixis block diagonal with uncoupled blocks. A typical element !1e generates a square diagonalblock of dimension dim(Ppe (!1e)) x m, namely, equal to the dimension of the space Ppe (!1e)

8

times the dimension of U. This means that, if necessary, this mass matrix can be invertedand stored at an extremely low cost, and the operation count of block diagonal matrix-vectorproducts is relatively low.

4.1 Algorithm for steady state computations

Let the restriction of the state vector Uh to any given element e be written as

where We is the matrix of local shape functions of dimension m x (Ppe (Oe)r, and Ue is thevector of element degrees of freedom of dimension (Ppe (Oe))m. Then, the algebraic systemof equations (10) can be written in matrix form as follows:

(11 )

here

N(Un) = L {{ W~ (Ft,,(U-) + F~e(U+)r dsn.ePh lan.

_ ( 8W; Fi(Un) dX} ,In. 8Xi

the mass matrix M is a block-diagonal matrix, each block having dimensions(dim(Ppe (Oe)))m x (dim(Pp. (Oe)))m, whereas ou, sn and N(Un) are vectors of dimensiondim(V p (Ph))'Given that M is a block-diagonal matrix, the system of equations (11) can be solved elementby element, as follows

(12)

here S: and N e(Un) are the components of sn and N(Un) associated with the test functionsWe which have local support on Oe.

9

4.2 Elementwise point-implicit scheme

The maximum allowable time step associated with any explicit time-marching scheme islimited by a CFL condition. To ameliorate this limitation, we use an elementwise point-implicit scheme.

The scheme is obtained by linearizing all those equations coming from weight functionswhose support is the domain of a given element, with respect to all the degrees of freedomassociated with the same element.

Using (12) with local linearization at element level, we get the following time-marchingscheme:

here

(1M as: aN: ) r (sn N ( n))8t e - QUe + aUe

uUe = e - e U(13)

and Me, s~,and N e are the same as before.

It is important to note that the scheme is applied element by element. For steady-statecalculations, the changes 8ue are used to update Ue and to compute the residual of otherelements as soon as tSue is obtained. In a Gauss-Seidel fashion, this updating accelerates theconvergence to the steady-state.

This relaxation scheme is only first order in time, but allows larger time-steps to reach thesteady state sooner (assuming that a steady state exist). Numerical experiments indicatethat a 8t equivalent to a CFL number of 2-5 is usually the optimum.

For time dependent problems, an explicit multistep Runge-Kutta could be applied.

4.3 Strategy for h-p adaptation:

From approximation theory, it is known that using high-order elements in zones where thesolution has low regularity is not optimal for linear problems, and the· situation may beworse for nonlinear problems. In fact, usually this is the origin of unstabilities in most

10

high-order schemes. Therefore, the usual strategy is to use low order elements (piecewiseconstant or piecewise linear with slope limiting or artificial diffusion) close to zones of sharpgradients, such as shock waves. In these zones, high resolution can only be obtained by usingh refinement. Numerical experiments indicate that using piecewise constant approximationsto resolve sharp layers which are oblique to the mesh is very diffusive, and that using piecewiselinear representations with artificial diffusion (only on elements close to the sharp gradients)leads to a better resolution. The artificial diffusion vector can be written as follows

here he is the characteristic size of element !1e, and Ae = Ae(U e) is the maximum character-istic speed, taken as c+ V, where c is the speed of sound and V the modulus of the velocityvector. This vector is scaled with an ad hoc parameter and is added to the RHS of (11). Letus note that the matrix 8D(U)/8u is a block-diagonal matrix, whose blocks are

8De (Ue) = _ [ heAe 8W; 8We dx,8ue Jne 8Xi 8Xi

and that the matrices De(Ue) can be used in the element by element algorithms describedby equations (12) and (13).

5 Numerical experiments

5.1 Accuracy Test:

This test is aimed at evaluating the accuracy of the DFEM applied to the Euler equations.To evaluate the h convergence rate (accuracy), we used nested grids obtained by successiverefinements of an initial grid covering the domain [0,3] x [0,3]. The RHS of (1) is specifiedso that the field variables p = 10+ x + y, u = 1+ y, v = 1+ x, and p = (1//)(10 + x + y)3,are the solution to the Euler equations.

Figure 3 shows the L2-norm of the error in density and the convergence rate. The asymptoticconvergence rate for p > 0 is clearly of order a (hp+1) , which is optimal. For the case p = 0,however, we observe a loss of order 1/2, a phenomena observed in other schemes [1].

5.2 Practical computations:

The first test case is the transonic flow around a NACA0012 profile at Moo = 0.80 with anincidence angle of 1.250

• The solution process is started on a C-mesh of 24 x 8 elements ofsecond and third degree, Fig. 4 shows the final mesh after adaptation, with the Mach number

11

distribution, on the background, and Figure 5 shows only the Mach number distribution.Figure 6 shows the pressure distribution, and the corresponding pressure coefficient distri-bution is shown in Fig. 7, in which the diamonds represent the classical GAMM solutionsobtained with structured finite difference techniques. The latter figure shows that the DGsolution with adaptation produces much sharper shock waves than typical finite differencetechniques with artificial diffusion.

The following test case is the transonic flow around a NACA0012 profile at Moo = 1.2 andincidence angle 7.00

• Figure 8 shows the final mesh after adaptation with the Mach numberdistribution, and Fig. 9 shows the pressure distribution. These figures show the advantageof using discontinuous approximations even in cases where the shock waves are oblique withrespect to the original mesh, in this test case the oblique shock wave that starts at thetrailing edge of the airfoil is very well resolved.

The next test case is a. standard benchmark for reflection and interaction of oblique shockwaves, it is a cascade with aspect ratio 4% and Moo = 1.40. Figure 10 shows the finalmesh and the Mach number distribution, and Fig. 11 shows the pressure distribution. Acomparison with finite difference solutions obtained using much finer gi'ids reveals that thesame quality of solution was obtained using our DG approximation with significantly lessnumber of elements.

5.2.1 Shuttle Orbiter at Moo = 7.40 and attack angle 400

The objective of this simulation is to compare the pressure coefficient from wind tunnelexperiments [10] and the values obtained with a numerical simulation.

The perfect gas model is used in the computations, this is justified because high-temperatureeffects were not present in the wind tunnel experiment.

Given that the aft section of the vehicle has been simplified, the comparisons with windtunnel data should be made only forward of the elevon hinge-line. This simplification isjustified because the flow in the aft region is predominantly supersonic, consequently, themodeling of the geometry past the elevon hinge-line has negligible upstream influence.

The solution is obtained using mesh sequencing, starting with a grid of 25 x 42 x 37 linearelements, executing the final iterations on a grid of 50 x 84 x 74 linear elements.

Figure 12 shows the streamline pattern, Fig. 13 the Mach number distribution at two normalplanes, and the Mach number and pressure distributions on the surface of the vehicle areshown in Fig. 14 and Fig. 15, respectively.

A comparison between the pressure coefficient from wind tunnel experiments [10] and thevalues obtained with the numerical simulations is shown in Fig. 16 for the centerline, and inFig. 17 for a cross section station z/ L = 0.6 (from nose to tail). The latter figure shows thepressure coefficient as a function of the a.ngle around the body.

12

These numerical results indicate that the dominant inviscid flow features are very well pre-dicted on the windward and leeside of the Shuttle Orbiter at a high angle of attack Asexpected, the aft portion of the vehicle is not well modeled past z/ L = 0.8 due to thegeometric simplifications introduced in the mesh past the elevon hinge-line.

6 Conclusions

The discontinuous Galerkin technique presented produces a compact, conservative, higher-order accurate, and stable solver. The compactness of the approximation permits the llseof different polynomial orders in each finite ele~ent, which can be adapted according tothe regularity of the solution, and makes possible a consistent implementation of boundaryconditions.

Regarding cost, the DGM may involve more degrees of freedom per element than standardFD /FV schemes, but the order of approximation can be varied adaptively and high-orderapproximations are achievable, the scheme generally requir~s fewer unk'nowns than conven-tional methods to achieve a target level of accuracy. Regarding the implementation on MPParchitectures, the attainable speedup using the DGM should be significant because the com-munication to computation ratio is very low for the DGM compared with standard FD /FVschemes. .

Several properties of the method deserve further comment:

• Ow\ng to the fact that the solution Uh and the approximate fluxes F(U h) can bediscontinuous across element boundaries, local element-based approximations can bedifferent in adjacent elements. For time-dependent approximations, the local (element)mass matrices are completely uncoupled. This means that the global mass matrices areblock diagonal with uncoupled blocks, even for high order approximations. This is aunique and useful property of high-order discontinuous finite element approximations.

• As noted previously, the method is elementwise conservative, a property not generallyexhibited by high-order FEMs.

• The method makes possible the construction of hp-FEM approximations with varyingand nonuniform distributions of the element spectral order Pe without the necessity ofboundary constraints common to most hp methods.

• The structure of this discontinuous Galerkin method, particularly the fact that thedegrees of freedom of an individual element are coupled only with those of neighborssharing a boundary, suggest that the method is easily parallelizable. Parallel algorithmsbased on the method are the subject of future research.

13

• In companion work, we have also developed similar versions of DGM for linear, second-order partial differential equations modeling diffusion phenomena [27]. The results in[27][2] and those presented in the present paper suggest that these techniques shouldbe effective in treating the full Navier-Stokes equations. We consider such extensionsin a forthcoming paper.

AcknowledgementThe support of this work by the Army Research Office under grant DAAH04-96-0062 isgreatfully acknowledged.

References

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14

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16

_ .....-.......~ ......

--..--------------- --- ------------ ......-----

.......--- ..-- ..-----

------~,~,~,

p=O -p:l m __

p:2 '"P=3~0.1

0.01

0.001

~ 0.0001

le-05

le.()6

1e-07

le.082 4

HIh8 •

5

-.----.--- ..- ....--.-- ....-----

p:O -P=l -----p=2 .....p--3 --.

4

CDiia:CDoc:CDe>~c:8

3

2

....... ---- --- .

..------.....---- ..-------------------------- ..

o2 4

HIh8

Figure 3: L2-norm of the error and convergence rate: 8F;j 8Xi = S,1 + y, v = 1 + x, and p = (1/1')(10 + x + y)3.

p = 10+x+y, u =

17

............ ,

,i/.",

\ ,\\\

_,T-I

Figure ·1: NACA0012, Moo = 0.80, incidellce 1.250, t-.1esh a.nd Mach number

: 1

Figure'!): NACAOOI2, AI",,) = 0.80, incidence 1.250 , Mach lIumber

18

,.-,..

Figllre 6: NACAOO12. Mce. = 0.80. illcidence 1.250, Pressure

0..o

1.21.1

10.90.80.70.60.50.40.30.20.1o __ n

.0.1

.0.2

.0.3

.0.4 ["'"

.0.5 .-- .

.0.6

.0.7·0.8.0.9

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Chord

Figure 7: NACAOOI2, M.x. = 0.80, incidence 1.250, Pressure coefficient.

19

Figure 8: NACA0012, MIX.' 1.20, incidence 7.00, IVlesh and Mach numher

Figure 9: NACA0012, MIX.. 1.20, incidence 7.00, Pressure

20

Figure 10: Cascade, aspect rat.io :1%. Mrx.. 1.40, rVlesh and ~lach numher

Figure 11: Cascade, aspect. ratio :1%, Mo:.. = 1.40, Pressure

21

Figure 12: Shuttle Orbiter, Moo = 7.'l, incidence 40.0° , Streamlines

Figure 13: Shutt.le Orbiter, Moc, = 7..1, incidence 40.0° , Mach number

22

Figure 14: Shut.tle Orbiter, Moo = 7.4, incidence 40.00 , Mach number

Figure] 5: Shuttle Orbit.er, 111"" = 7.4. incidence 40.001 Pressure

2 , iii iii

0.8

Experiment: Test OH38 0Approximate: Euler -.-.-

0.60.4zA.

o~_._._._.__.__.4-'-'-'-'-'-'-'-'-'-'-'-'-'-'--'-'-'-'- '-\-'.

' ..

0.2

~iij\;\i ii.I \i \i \I \i \j "'i \i '.I '. 0I "i "• '" <>! " ...i· ..·...j;iIii;j;

tiiiii 0

\·, ....._._.~._9_..,,/·-·,o..._._._._ ... __ ._._._._._._._._.+._._. __ ._+_._,--_._._ ...o-._.~-._.-.-._'-'-'-'-'-', , I I·.

o

o

1.5

0.5

Q.

U

Figure 16: Pressure coefficient along the windward and leeward centerlines

I I

Experiment: Test OH38 0Approximate: Euler -.-.-

II

...._._._._.!_._ .-' - .'0 ;-.'.\\iiiii;;iiiiiiijiii;iiIjjiiiiiiiiI;i\ .....~._. _.!It. __ .~_.~-""_.-+--.---.-.-;;' .......-._._._._._._._._._._._._._._.__.o

0.2

0.4

0.8

0.6

Q.U

o,

20 40 60....L.

80 100Angle

120 140 160 180

Figure 17: Pressure coefficient at the fuselage station Z/ L = 0.6.

24