stable implementation of boundary conditions for dgsem

U⇤ V ⇤ W⇤ P⇤ ⇤. 2. 6. 6. 6. 6. 4. 0 т1 т2 т3 r. ∂P. ∂n. 3. 7. 7. 7. 7. 5. ≤ 0,. Vanishes if. Isothermal. Adiabatic x i. * =x i. U* =V* =W* =0. +. P* =0. ∂P*. ∂n. = ∂P.
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STABLE IMPLEMENTATION OF BOUNDARY CONDITIONS FOR DGSEM APPROXIMATION OF THE COMPRESSIBLE EULER AND NAVIER-STOKES EQUATIONS David A. Kopriva Florida State University and San Diego State University Gregor J. Gassner, U. Cologne and Andrew R. Winters University of Cologne Florian Hindenlang Max Planck Institute

A Flow Geometry

A Compressible Flow

Another Compressible Flow

Boundary Conditions Determine the Flow! Yet: • Hardly discussed except in passing • Often dealt with in ad-hoc manner • Published proposals not stable

We study… • Conditions under which DGSEMs are stable • Examples of stable BC implementations • General Analysis • 3D • Curved Elements • Linear and nonlinear equations

1. Introduction

G.J. GASSNER, A. R. WINTERS, F. HINDENLANG AND D.A. KOPRIVA

Compressible Flow Model 2. The DGSEM for Compressible Viscous Flows 1. Introduction

Navier-Stokes Equations: Conservative formequations, essible viscous flows are modeled by the Navier-Stokes 2. The DGSEM for Compressible Viscous Flows

3 3 X X @fi 1 @fv,i (u, rx u) ut + = . Compressible viscous flows are modeled by the Re Navier-Stokes equations, @x @x i i i=1 i=1

vector contains the

3 3 X X @fi 1 @fv,i (u, rx u) conservative variables ut + = . @x Re @x 2 3 i i i=1 i=1

Conservative Variables 2

⇢ 3 e state vector contains the conservative variables 3 ⇢u ⇢ 2 6 6 ⇢6 2 3 4 5 u = ⇢ ⇢~v 6 =⇢u67 ⇢v 6 7 4 5 = 6 ⇢v 7.⇢w u = 4 ⇢~v ⇢E 6 7 4 ⇢w 5 ⇢E ⇢E ⇢E

rd form,form, the the components ofthe theadvective advective flux are tandard components of flux are 22 32 33 2 2 3 ⇢u ⇢u ⇢v ⇢v

7 7 7. 7 5 ⇢w

2

3

⇢w

3

the divergence of a flux is defined as

his notation the divergence of aX 3flux is defined as $ @f i ~ 3 rx · f = , X $ @f i @x ~ i rx · f = , i=1 @x i HINDENLANG 2 G.J. GASSNER, A. R. WINTERS, F. i=1

Compact Version

ite the Navier-Stokes equations compactly as ows us to write the Navier-Stokes equations compactly as ⇣ ⌘ $ 1 ~will $simplify For a compact ~notation that the analysis, we d ⇣ ⌘ ~ $u |. |ut + rx · f = ~ r$x · fv1 u, r x ~ ~ x u |. |ut + Re rx · f = rx · fv u, r arrow) Re

2

3

f 1 $ roximation procedure, it is customary to represent the so 4 5 Vector of Vectors: f (8) | f = | t of the approximation procedure, it is customary to 2 represent firsttoorder of system equations f3 ble get a system first order of equations $ $ 1 $ $$ 1 $ and the Write spatialas gradient of a state as ~ ~ ~ ~ 1st order system with u t + rx · u f t=+ rx r · f v (u, · fx = rxq) · f v (u, q) 2 3 Re Re u x $ $ ~ ~ | q = r u|. q = r u. x x ~ u = 4 u 5. (9) r x

y

uz up the standard spectral element approximation, one subdivid

DGSEM Approximation Subdivide domain into elements

Curved Elements OK!

Unstructured Grids OK

Unstructured Grids OK

DGSEM Approximation Map Element to Reference Element

E 1

Isoparametric Map -10

e

1

unit vectors. Similarly, thereference reference element represented ⇠ = ⇠⇠ + ⌘ ⌘ˆ in computational space on the element E = [space 1, 1]3 .isEach element is by mapped

⇣ ⌘ ~ ⇠~ , where X ~ = Xx reference element with a mapping ~x = X ˆ + Y yˆ + Z x ˆ and the hats he transformation, we define the three covariant basis vectors ˆ unit vectors. Similarly, the reference element space is represented by ⇠~ = ⇠ ⇠ˆ + ⌘ ⌘ˆ + ⇣ ⇣.

Transformation of Operators

~ @X |~ai =covariant ibasis = 1,vectors 2, 3|, he transformation, we define the three i @⇠

me weighted)

~ @X |~ai = vectors, i = 1, 2, 3|, contravariant formally i @⇠

written as

me weighted) contravariant vectors, |J ~ai =formally ~aj ⇥ ~ajwritten , (i, as j, k)

cyclic|,

|J ~ai = ~aj ⇥ ~aj , (i, j, k) cyclic|, G.J. GASSNER, A. R. WINTERS, F. HINDENLANG A

Gradient 4 2

3

2

1 $J a1 I5 4 J a12 I5 J a13 I5

J a21 I5 J a22 I5 J a23 I5

J a31 I5 J a32 I5 J a33 I5

32

3

ux u⇠ 1 g, 1 so r a block vector ~for ~ ⇠u 4 5 5 4 5 uy u⌘ = = Mr xu = J J uz u⇣ 1 $ R, A. R. WINTERS, F. HINDENLANG AND D.A. KOPRIVA T$ ~ ~

(24)

| rx · g =

Divergence

J

r⇠ · M g |.

Finally, if we define the contravariant block vector Define Contravariant fluxes:

1 T$ ~ ~ |rx · g = r⇠ · M g |. J (25) $

$

$

|˜f = M f |, T

Finally, if we define the contravariant block vector nt of a state as 3 ctors is defined by 2 $ $ $ $ ux T ˜f = T f, ˜ 3 (25) = M (25) | f M f |, X $ $~ 4 5 u r u = . T f · g =x fi gi . y uz equations are compactly written as i=1 the Navier-Stokes

NS in Reference Coordinates vectors is defined by the Navier-Stokes equations are compactly written as

ck vector with a vector is a state vector, 3 3 X X $ (26) ~g · ff = (26) · g = gi fi . fi T gi . $ $

i=1

i=1

$$ 1 1 $ $ ~ ~ ˜ ˜ ~ ~ |Juutt + +r r⇠⇠ ··ff == r r⇠⇠· ·ffvv(u, (u,q)| q)| |J Re Re $ $ ~⇠⇠u| ~r |Jqq = =M Mr u| |J $ $

block vector with a vector is a state vector,

ence of on a flux defined as element. the on theisreference $3

3 X

X @fi gi fi . ~ g · f = ~ rx · f = , $

E

1 The spectral element approximation isis derived The approximation derived from from weak weakforms formsofo @x i=1i i=1 define the inner product on define on the the reference reference element element for forstate statevectors vectors ier-Stokes equations compactly as ZZ 11 ⇣ ⌘ ergence of a flux is defined as -1 $ 0 1 ~ $ TT ~ ~ ut + rx(27) ·f = rx · fv3 u, rx u |. (27) hv, wi = u hv, wi1EE = u vd⇠d⌘d⇣. vd⇠d⌘d⇣. e Re $ X

~x·f = r

@fi , @xi

11

procedure, it is customary to represent Similarly, fori=1 block vectors, Similarly, vectors, the solution gradients as a system of equations Navier-Stokes equations compactly as ZZ 11 X 33 D E D E ⇣ ⌘ $ $ $ X 1 1~ $ $ $ $$ TT ~ $ u + r · f = r · f (u, q) ~ ~ ~ t x x v (28) f , g = f |ut +(28) rx · f =Re rx · fv u, rx u |. f, g E = fi i ggi d⇠d⌘d⇣. i d⇠d⌘d⇣. Re 11i=1 E $ ~ x u|. i=1 |q = r

~ ⇠ u| Re |J q =$$ Mr $ $ $ ~ ⇠ u| T T |J q =˜˜ M r f = M f,

6)

|f = M f |, the reference element. the reference element. The spectral element approximation is derivedasfrom equations are written as weak forms of the equations (26). L equations are compactly compactly written Thethe spectral derived for from weak forms of the equations (26). L fine inner element product approximation on the referenceiselement state vectors $$ $ $ fine the inner product on the reference element vectors Z111for state $ $ ˜ ˜ ~ ~ ~ ~ T ·⇠ f·vf(u, q)|q)| |J u utt + +r r ·f fE== 7) |hv, =Z 1r ur ⇠⇠·wi ⇠vd⇠d⌘d⇣|. v (u, 1 uT vd⇠d⌘d⇣|. Re 7) |hv, wiE = Re $ 1 $ ~ ~⇠⇠u|u| milarly, for block vectors, |J q |J q= =M Mrr milarly, for block vectors, Z 1X 3 D$ E $ 3 fiT gi d⇠d⌘d⇣|. lement. 8) | D$ f , gE = Z 1 X ement. (1) Take inner product $ of equations with test functions 8) | f , g E = Z 1 i=1 fiT gi d⇠d⌘d⇣|.

Weak Form Construction

E

1 i=1

vi = we uvdE nce there should be no confusion inderived context, will usually leave o↵ the subscript E. The ement approximation ishu, from weak forms of the equation E nce there should be no confusion context, we from will usually leaveforms o↵bythe subscript E.equa The rms that serve as the starting point the approximation are created multiplying equ ment approximation isinofderived weak of theeach roduct on as the for vectors rms serve thereference starting of the approximation are created by multiplying equ an that appropriate test functionpoint andelement integrating overstate the element. After integration each by parts (2) Apply Gauss Law oduct on the reference element for state vectors anform appropriate test function and integrating eak of (26) reads as Z 1 over the element. After integration by parts eak form of (26) reads asZ ⇢$ D$ E D$ E $Z 1 T 1 1 T ⇢˜ = ˜ vd⇠d⌘d⇣. ˜$ ˜$ ~⇠ E= ~ E| |hJ u, i + Zhv, wi f$ f$v ·u n ˆ dS f, r f v, r E D D T 1 ˜ 1 u vd⇠d⌘d⇣. 1 ˜ ~ Re Re T hv, wi = @E ˜ ˜ ~ |hJ fv · n ˆ dS D f , r⇣ = ⌘E f v, r | 9) ⇠ D u, $iE+ Z nfE Re o Re $ $ @E $ 1 T T T ~ · ⇣M ⌘E| 9) | DJ q, $E = Z u nM $o · n ˆ dS Du, r $ k vectors,| J q, $ T T T @E ~ · M = u M ·n ˆ dS u, r |

vectors,

@E

D$ E

Z

1

3 X

) ilarly, for block vectors, ilarly, for block vectors,

|hv, wiE = Z |hv, wiE =

1 1

uT vd⇠d⌘d⇣|. uT vd⇠d⌘d⇣|.

1

Approximate Z D$ E $ |D$ f , gE = Z $ | f, g E =

)

E

3 X 3 fiT gi d⇠d⌘d⇣|. 1 X 1 i=1 f T g d⇠d⌘d⇣|. i i

1

1 i=1

Functions with polynomials ce there should be no confusion in context, we will usually leave o↵ the subscript E. The w ce there should be no confusion we will usually leave o↵bythe subscript each E. The w ms that serve as the starting pointinofcontext, the approximation are created multiplying equat Boundary quantities ms serve as test the starting the approximation are created by multiplying equat an that appropriate functionpoint and of integrating over the element. After integration each by parts, an appropriate test function and integrating over the element. After integration by parts, k form of (26) reads as with numerical ones k form of (26) reads asZ ⇢$ D$ E D$ E $ 1 1 T ⇢˜ ˜f v · n ˜f , r ˜f v , r ~⇠ E= ~ E| |hJ u, i + Z f ˆ dS D$ D $ $ $ 1˜ 1 ˜ ~ Re Re T ˜ @E ˜ ~ |hJ fv · n ˆ dS D f , r⇠⇣ = ⌘E f v, r | D u, $iE+ Z nf Re o $ $ Re @E $ T T T ~ · ⇣M ⌘E| ) | DJ q, $E = Z u nM $o · n ˆ dS Du, r $ $ T T T @E ~ · M | J q, = u M ·n ˆ dS u, r | @E

The Spectral Element Approximation. To get spectral accuracy, we approximate eThe vector by polynomials of Approximation. degree N , which we as Uaccuracy, 2 PN (E). polynom Spectral Element Torepresent get spectral we The approximate N terms of the Lagra Integrals written terms of the functions, or equivalently ebevector by inpolynomials of Legendre degree Nbasis , which we represent as U 2 Pin (E). The polynom sbe with nodesinwith atterms the Legendre Gauss orbasis Gauss-Lobatto points with nodal values of Unml n, m, written of the Legendre functions, or equivalently in terms the ,Lagra quadrature N . , N .nodes We write interpolation function g through nodes values as G =UInml (g). s. .with at thethe Legendre Gauss of or aGauss-Lobatto pointsthose with nodal , n,Flu m, also approximated with polynomials degree N represented fr ...,N . We write the interpolation of of a function g ,through thosenodally, nodes asand G =computed IN (g). Flu

Continuous Function Approximation

E 1

-10

Approximation by Polynomial Interpolant

e

1

N ⇣ ⌘ X U ⇠~ = IN (u) = u (⇠i , ⌘j , ⇣k ) `i (⇠) `j (⌘) `k (⇣) i,j,k=0

`j (x) =

N Y

i=0;i6=j

x x

xi = Lagrange Interpolating Polynomial of degree N xj xj = Gauss Lobatto points

`j (xi ) =

ij

Arbitrary High Order Acoustic Scattering from a Cylinder

74th Order

Differentiation Differentiate interpolant, evaluate at quadrature points @U @⇠

= nml

N X

uijk `0 i (⇠n ) `j (⌘m ) `k (⇣l )

i,j,k=0

=

N X i=0

0

uijk ` i (⇠n ) =

N X i=0

uijk Dni

Differentiation Gradient rUijk =

N X

n=0

Unjk Din ⇠ˆ +

N X

n=0

Uink Djn ⌘ˆ +

N X

n=0

Uijn Dkn ⇣ˆ

Divergence r · F~ijk =

N X

n=0

(⇠) Fnjk Din ⇠

+

N X

n=0

(⌘) Fink Djn

+

N X

n=0

(⇣) Fijn Dkn

Integral Approximation E

1

Gauss-Lobatto Quadrature

-10

e

Integration over Volume Z

E,N

gd⇠d⌘d⇣ ⌘

N X

gijk wijk ,

i,j,k=0

(wijk = wi wj wk )

Defines discrete inner product/Norm hU, V iN ⌘

Z

U V d⇠d⌘d⇣ = E,N

N X

i,j,k=0

Uijk Vijk wijk

1

Summation-By-Parts Exactness of Gauss Quadrature implies Integration By Parts (u, v⇠ ) =

Z

UV @E

1 |⇠= 1

d⌘d⇣

(u⇠ , v)

+

Summation by Parts Z 1 (U, V⇠ )N = U V |⇠= 1 d⌘d⇣ @E,N

(U⇠ , V )N

Summation by Parts works in each direction hU⇠ , V iN =

Z

hU⌘ , V iN =

Z

F1

F2

hU⇣ , V iN = F3

UV

1 |⇠= 1

d⌘d⇣

hU, V⇠ iN

UV

d⇠d⇣

N

1 |⌘= 1

hU, V⌘ iN

UV

d⇠d⇣

N

1 |⇣= 1

hU, V⇣ iN

N

Z

Discrete Gauss Law ⇣

r · F~ ,



N

=

Z

@E,N

F~ · n ˆ dS



F~ , r



N

Discrete Integral Calculus ⇣

Z r2 , V r2 , V

N

r · F~ ,

E,N



N

Z

=

r · F~ d⇠~ =

@E,N

Z

@E,N

+ (r , rV )N = r2 V,

N

N

=

Z Z

F~ · n ˆ dS



F~ , r



N

F~ · n ˆ dS

@E,N

r ·n ˆ V dS

@E,N

(r · n ˆV

rV · n ˆ ) dS

From Exactness Z

E,N

rV d⇠d⌘d⇣ =

Z

Vn ˆ dS @E,N

Z

E,N

~ d⇠d⌘d⇣ = r⇥F

Z

@E,N

~ dS n ˆ⇥F

Coupling-Advective Riemann Solver F⇤ U L , UR

! ! L f U ·n ˆ + f UL · n ˆ = + A¯ UL 2 2 nn! oo = f ·n ˆ

2

A¯ [[U]]

Roe Lax-Friedrichs van Leer …

UR

⇢ Z E D$ E mpressible Navier-Stokes equations D $ 1 1 T ⇢ ⇤ ˜$ r ˜$v , r ~⇠ E = ~⇠ E | +Z F F⇤v dS DF, F D 1 ⇤ 1 ˜ ~ Re Re N N T ⇤ @E,N ˜ ~ + F F dS F, r = F , r | ⇠D v ⇠ Z v N ⇣ ⌘ E Re Re N N $ $ @E,N T ⇤⇤ T T ~ ⇠ U, M = Z {U U} ⇣M $⌘ · n ˆ dS Dr E |, $ N T ~ ⇠ U, MT N |, = @E,N {U⇤⇤ U} MT ·n ˆ dS r

N

@E,N

Coupling - Diffusive N

vective flux F⇤ is usually computed with an approximate Riemann solver such vective flux F⇤ is usually computed with anfor approximate Riemann solver such s or Roe solvers. The coupling functions the viscous terms include the hsBassi-Rebay-2 or Roe solvers. TheInterior coupling functions theothers. viscousThe terms include the (BR2), Penalty (IP),forand simplest is the Bassi-Rebay-1: Bassi-Rebay-2 (BR2), Interior Penalty (IP), and others. The simplest is the chooses chooses L R U + U |U⇤⇤ UL , UR = UL + UR = {{U}}| Others… |U⇤⇤⇣ $UL , U$R = ⌘ 2 $ = {{U}}| 2˜ ⇤ ˜L R ˜ |Fv ⇣F ˆ, F ˆ ⌘ = {{F ˆ }}|. $v · n $v · n $v · n Bassi-Rebay-2 ⇤ ˜L R ˜ ˜ |Fv Fv · n ˆ , Fv · n ˆ = {{Fv · n ˆ }}|.

Interior Penalty n with an upwind Riemann solver for the advective flux and the BR1 scheme …include onis with an stable upwindinRiemann advective flux andExamples the BR1 scheme usually practice, solver at leastforforthe well-resolved flows.

snsional is usually stable in practice, at least for well-resolved include computations, e.g. [?],[?],[?]. Often, however,flows. someExamples kind of filtering nsional e.g. integrals [?],[?],[?].areOften, however, some of filtering stabilitycomputations, [?] or the volume “overintegrated”, i.e.,kind evaluated with stability [?],[?]. [?] or the volume integrals are “overintegrated”, i.e., evaluated with [?],[?].

)

|hv, wiE =

u vd⇠d⌘d⇣|. 1

he spectral element approximation is derived from weak forms of the equations (26). Let ilarly, for block vectors, ne the inner product on the reference element for state vectors Z Z1 X 3 D$ E 1 $ T ) | |hv, f , gwi == Tfi gi d⇠d⌘d⇣|. u vd⇠d⌘d⇣|.

Weak Form Construction… EE

1 i=1 1

ce there be no confusion in context, we will usually leave o↵ the subscript E. The w larly, forshould blockGauss vectors, Apply Law again… ms that serve as the starting point of the approximation are created by multiplying each equat Z 1 X 3 D E $ h·, ·iintegration an appropriate test function and $integrating by parts, N overTthe element.⇡After N N | f, g 2 = fi gi d⇠d⌘d⇣|. P ⇡ U 2 P k form of (26) reads as E 1 i=1 ⇢$ $ Z D E D$ E $ $⇤ 1 1 T e there |hJ should we usually leave E. |The we ˜f , o↵=the subscript ˜f v , r ˜ ⇤ in ˜context, ˜ willdS ~ ~ ut , be i +no confusion F f ·n ˆ F r · ⇠ v Re Re ms that serve as the starting point of the approximation are created by multiplying each equat @E ) Z D E n o the element. D ⇣ ⌘E $test function and $ $ n appropriate integrating over After integration by parts, Z $ T T ⇤⇤,T ~ | J q, = U u M ·n ˆ dS u, r · M | k form of (26) reads @E N as # ⇡ 2P Z ⇡ D F ⇢$ $ D E D$ E $ $⇤ 1 1 @E,N T ˜f , ˜f v , r ˜ ⇤ ˜f · n ˜ ~ ~ |hJ u, i + F ˆ F dS r · = | ⇠ v Re Re we approximate @E The Spectral Element Approximation. To get spectral accuracy, D E Z n o D ⇣ ⌘E $ $ $N $ polynomials of⇤⇤,T T e vector byq, which we The polynom ~as· UM2T P (E). | J = U degreeu N ,M ·n ˆ represent dS u, r | be written in terms @E of the Legendre basis functions, or equivalently in terms of the Lagra s with nodes at the Legendre Gauss or Gauss-Lobatto points with nodal values Unml , n, m, . . . , N . We write theNinterpolation of a function g through those nodes as G = IN (g). Flu ⇡ Q Element 2 Pwith polynomials The approximated Spectral Approximation. ToNget spectral accuracy, also of degree , represented nodally, we andapproximate computed fr N enodal vectorvalues by polynomials of degree N , which we represent as U 2 P Thedi↵erentiat polynomi of the state and gradients. Derivatives are approximated (E). by exact

???

Split Form/Two Point Flux ~ f~ = A(x)u n o 1 ~ · ru + r · Au ~ r · f~ + A Split Form: r · f~ = 2 1 2



⌘ ! r · F (U) ,

= `i `j `k

N

⇣ ⇣ ⌘ ⌘ ⌘ 1⇣ ⇣ ⌘ 1 N N ~ U, + ~ + r · I A I A · rU, 2 2 N N

Volume Terms ⇣ ⇣



! r · F (U) ,

wijk ⇣

(

N X

(⇠) Fnjk Din

Unjk Din +

N X

wijk

N

⇣ ⌘ ⌘ N ~ · rU, I A

n=0

+

N X

n=0

(⌘) Fink Djn

+

N X

n=0

(⇣) Fijn Dkn

)

N

(⇠) Aijk

N X

n=0

⇣ ⌘ ⌘ ~ U, r · IN A N ( wijk

(

N X

n=0

(⇠) Anjk Din

+

(⌘) Aijk

N X

n=0

n=0

Uink Djn +

(⌘) Aink Djn

+

(⇣) Aijk

(⇣) Aijk

N X

n=0

N X

n=0

Uijn Dkn

(⇣) Aijn Dkn

)

)

Uijk

Volume Terms ⇣ 1 2

! r · F (U) ,



N



⇣ ⌘ ⌘ ⇣ ⇣ ⌘ ⌘ 1 N ~ 1 N ~ U, I A · rU, r · I A + + N 2 N 2

N n o X 1 (⇠) (⇠) (⇠) Fnjk + Aijk Unjk + Anjk Uijk Din wijk 2 n=0

N n o X 1 (⌘) (⌘) (⌘) + Fink + Aijk Uink + Aink Uijk Djn wijk 2 n=0

N n o X 1 (⇣) (⇣) (⇣) + Fnjk + Aijk Unjk + Anjk Uijk Dkn wijk 2 n=0

=

N X

n=0

(⇠) ¯ F(njk,i) Din wijk +

N X

n=0

(⌘) ¯ F(ink,j) Djn wijk +

N X

n=0

(⇣) ¯ F(ijn,k) Dkn wijk

Special Averages 1 2 1 2

N n X

(⇠) Fnjk

+

n=0

(⇠) Aijk Uijk

(⇠) Aijk Unjk

+

(⇠) Anjk Uijk

+ N X

n=0

Din =

N X

n=0

o

Din

(⇠) Aijk Uijk Din

=0

=0 N n o X 1 (⇠) (⇠) (⇠) (⇠) = Anjk Unjk + Aijk Unjk + Anjk Uijk + Aijk Uijk Din 2 n=0

Special Averages N n o X 1 (⇠) (⇠) (⇠) (⇠) = Anjk Unjk + Aijk Unjk + Anjk Uijk + Aijk Uijk Din 2 n=0 N X

=2

n=0

=2

(

N nn X

n=0

=2

N X

n=0

(⇠) Anjk

A(⇠)

+ 2

oo

(⇠) Aijk

(n,i)jk

# F(n,i)jk Din

!✓

Unjk + Uijk 2

{{U}}(n,i)jk Din

◆)

Din

Key Ingredient #

Summation-By-Parts and form of F implies

! 1 # T ˆ DF ,U = ∫ U F⋅ ndS 2 ∂E ,N Volume term replaced by surface quadrature Control with numerical flux

rly, for block vectors, (27)

|hv, wiE = uT vd⇠d⌘d⇣|. 1 Z 3 1 X D$ E $ Similarly, for block vectors, | f, g = fiT gi d⇠d⌘d⇣|. E 1 i=1 Z 1 X 3 D$ E $ (28) should be no confusion in context, | f , g we=will usually fiT gleave there o↵ the subscript E. The wea i d⇠d⌘d⇣|. E 1 i=1 that serve as the starting point of the approximation are created by multiplying each equatio appropriate function integrating over the element. After integration by parts, Since there test should be no and confusion in context, we will usually leave o↵ the subscript E. th T form of that (26) serve readsas asthe starting point of the approximation are created by multiplying each forms ⇢ $ $ and integrating Z ⇣ $element. ⌘ E After 1integration D$ Eby p by an appropriate test function overD the $⇤ 1 T # ˜ ⇤ ˜f · n ˜ ˜ v, r ~ |hJ U , i + ˆ F dS D F , = F | weak tformN of (26) reads asF v Re Re N N @E,N ⇢ $n $ o Z Z D ⇣⇣ $ ⌘ D$ D $ $E D ⌘EE $⇤ $ $ 1 1 # ⇤ ⇤⇤,T T T· n T ˜ ˜ ˜ ˜ v, r ~ ~ , i + F f ˆ F dS D F , = F | J Q, |hJ U= U U M · n ˆ dS U, r · M | t v N Re Re N NN @E,N @E,N Z (29) D $ $E n o D ⇣ ⌘E $ $ ~ · MT | J Q, = U⇤⇤,T U MT ·n ˆ dS U, r |

DGSEM

N

@E,N

N

The Spectral Element Approximation. To get spectral accuracy, we approximate th vector by polynomials of degree N , which we represent as U 2 PN (E). The polynomia e 2.1. written terms of the Legendre basis functions, or terms of the Lagrang Thein Spectral Element Approximation. Toequivalently get spectralinaccuracy, we approxim with nodes at the Gauss or Gauss-Lobatto points with nodal Unml ,The n, m, l= state vector by Legendre polynomials of degree N , which we represent as U values 2 PN (E). pol N . ,can N . be Wewritten write the interpolation of a function g through those nodes as G = I (g).of Fluxe in terms of the Legendre basis functions, or equivalently in terms the L sobasis approximated of degree N , represented nodally, and computed from with nodes with at thepolynomials Legendre Gauss or Gauss-Lobatto points with nodal values Unml , odal the write state and gradients. Derivatives are approximated bynodes exact as di↵erentiatio 0, 1,values . . . , Nof. We the interpolation of a function g through those G = IN (g) polynomial interpolants.with Di↵erentiation and interpolation do not commute, however, s are also approximated polynomials of degree N , represented nodally, and compu 0 N ) the 6= Inodal (g 0 )[?],[?]. values of the state and gradients. Derivatives are approximated by exact di↵er

| f, g

(28)

D= E fi 1gX i d⇠d⌘d⇣|. $ $ E | f , g 1 i=1 = fiT gi d⇠d⌘d⇣|. E

1 i=1

here should be no confusion in context, we will usually leave o↵ the subscript E. The weak Since there should be nopoint confusion context, we will usually by leave o↵ the subscript E. T hat serve as the starting of the in approximation are created multiplying each equation forms that serve as the starting point of theover approximation created by multiplying each e appropriate test function and integrating the element.are After integration by parts, the by an appropriate orm of (26) reads astest function and integrating over the element. After integration by pa weak form of (26) Z reads⇢as$ $ D ⇣$ ⌘ E D$ E $⇤ 1 1 #D ⇣ ZT F ˜ ⇤ ˜f⇢·$n ˜ ˜ v, r ~ D$ | ⌘ E $⇤ |hJ Ut , iN + ˆ $ F dS D F , = F $ v 1 ˜ 1 ˜N ~ T # ⇤ Re Re N ˜ ˜ @E,N |hJ Ut , Z iN + F f ·n ˆ Fv dS D F , = Fv , r Re D $ $E n o Re D ⇣ ⌘E N @E,N $ $ T ⇣ ZU⇤⇤,T U MT n · n ~ o D |(29) J Q, D $= $E ˆ dS U, r · M | $⌘E $ NT ~ · M | JNQ, @E,N= U⇤⇤,T U MT ·n ˆ dS U, r |

Stability Analysis

N

N

@E,N

he Spectral Element Approximation. To get spectral accuracy, we approximate the 2.1. The Spectral Element To get approxim ector by polynomials of degree Approximation. N , which we represent as spectral U 2 PNaccuracy, (E). Thewe polynomial N Linear Entropy state vector by polynomials of degree N , which we represent as U 2 P (E). poly written in terms of the Legendre basis functions, or equivalently in terms of the The Lagrange cannodes be written terms ofGauss the Legendre basis functions, or equivalently in U terms of the L with at thein Legendre or Gauss-Lobatto points with nodal values nml , n, m, l = withwrite nodes the LegendreofGauss or Gauss-Lobatto points withas nodal nml , n ,basis N . We theatinterpolation a function g through those nodes G =values IN (g).UFluxe N (entropy variable) 0, 1, . . . , N . We write the interpolation of a function g through those nodes as G = I (g). • Linearize equations o approximated with polynomials of degree N , represented nodally, and computed from T arevalues also approximated with polynomials of degree N , representedbynodally, and comput −1 −1Derivatives dal of the state and gradients. are approximated exact di↵erentiation • Replace φ ←W φ = S S U • Replace the nodal values of the state and gradients. are do approximated by exact di↵ere polynomial interpolants. Di↵erentiation andDerivatives interpolation not commute, however, so 0 the of interpolants. Di↵erentiation and interpolationψdo commute, how ←not B∇W N polynomial 0 ψ ← BQ 6= I (g )[?],[?]. 0 N I (g) 6= IN (g 0 )[?],[?].

( )

• Sum over all elements

• Sum over all elements

BRIEF ARTICLE BRIEF

ARTICLE

Linear Energy Bound THE AUTHOR

d dt

Z ARTICLE X X BRIEF 2 ||U||J,N 

2

(

THE AUTHOR

⌘ 1 ⇣$ ˜ ·n F ˆ 2 Z

˜⇤ F

T

⇣$ ⌘ 1 h ˜ ⇤,T ˜v · n U( Fv U + U⇤,T F ˆ Re ⇣$ ⌘ T

X dTHE AUTHOR 2 ||U||  J,N h 2 T ⇣ ⌘ ⇣$ ⌘ $ 1 ⇤ dt 1 @E,N ⇤,T ˜ ˜ ˜ ˜ ⇤,T Boundary

elements

X

elements F·n ˆ

F

2

(

@E,N

Boundary faces

U+

Re

Fv U + U

faces

Fv · n ˆ





˜v · n UT F ˆ

SufficientF˜ ⇤ConditionU for Stability: X Z

2

Boundary faces

h

@E,N



⇤ ˜ F

˜ ⇤,T U + U⇤,T F v

⌘ 1 ⇣$ ˜ ·n F ˆ 2

⌘ 1 ⇣$ ˜ ·n F ˆ 2 ⇣$ ⌘ ˜v · n F ˆ

T

T

1 U − Re



$

◆T

⇣$

1$ ˜ ⌘ ⇤ ⇣ ˜ 1 h ˜ ⇤,T ˆ ˜ vF· n Fv U +FU⇤,T 2 F ˆ· n Re

h

˜ ⇤,T U + U⇤,T F v

⇣$ ⌘i ˜v · n UT F ˆ

• 3D ˜⇤ = F ˜v · n F v • Curved Hexˆ Elements ˜⇤ = 0 F • Any Polynomial Order v $

˜ F ⇣$

1 ˜ F·n ˆ ⌘i 2

⇣$ ⌘ ˜v · n F ˆ

U

0.



T

⇣$ ⌘i T ˜ UU Fv · n ˆ

⇣$ ⌘i ˜v · n UT F ˆ

)

dS

⇣ 1 h ˜ ⇤,T Fv U + U⇤,T Re

)

⇣$ ⌘i ˜v · n UT F ˆ

dS

≥0

N

BRIEF ARTICLE

X Z

(

BC Implementation d X 2 ||U|| THE AUTHOR J,N  dt

2

elements

Boundary faces

@E,N



˜ F



⌘ 1 ⇣$ ˜ ·n F ˆ 2

T

U

1 h ˜ ⇤,T Fv U + U ⇤ Re

⇣$ ⌘ T ) 1 Z T ⇣ ⌘ h ⌘ ⇣ ⌘i $ $ ˜ ˜ ·n X F F ˆ U 1 1 ⇤ ⇤,T ⇤,T T ˜ ˜ ˜ ˜ ˜ Implementations  BC 2 F F·n ˆ U are Fstable U + U ifFv 2· n ˆ U Fv · n ˆ dS 2 Re v @E,N

Boundary faces



(SSC) h

˜ ⇤,T U F v

(

⌘ 1 ⇣$ ˜ ·n F ˆ 2

˜⇤ F

+U

⇤,T

⇣$ ⌘ ˜v · n F ˆ

T

1 U − Re

h

⇣$ ⌘i ˜v · n U F ˆ T

Dirichlet-Type ˜⇤ F v

$

˜v · n =F ˆ

⇤⇣ $

˜ ⇤,T U + U⇤,T F v

⇣$ ⌘ ˜v · n F ˆ

⇤ ˜ F

✓ ⇢

˜ F



$

1˜ F·n ˆ 2 ˜⇤ F

◆T

U

1$ ˜ ·n F ˆ 2

⌘ 1 ⇣$ ˜ ·n F ˆ 2

◆T

T

U

1 T + U A U 2

0

Robin-Type Conditions

U = aN2

0

⌘o 1 n T ⇣$ ˜v · n U F ˆ Re

≥0

Neumann-Type

˜⇤ = 0 F v ✓

⇣$ ⌘i ˜v · n UT F ˆ

THE AUTHOR

(

Typical Implementation

X Z

2

Boundary 2 faces

||U||J,N 

@E,NX

2

⌘ T h ⇣$ ⌘ 1 ⇣$ 1 ˜ ·n ˜ ⇤,T U + U⇤,T F ˜v · n F ˆ U F ˆ v ( 2  Re T ⇣ ⌘ ⇣$ $ 1 1 h ⇤

⇤ ˜ F

Z

˜ F

2 Boundary @E,N T ⇣ ⌘ $ faces 1 ⇤ ˜ ˜ ·n F ˆ ⇣ U ≥⌘0T F 2 1 $ ⇤

(E)

˜ F

h

˜ ·n F ˆ

˜ ⇤,T U +hU⇤,T F v

(D)

2

˜ ·n F ˆ

˜⇤ = F ˜v · n F ˆ v ˜⇤ = 0 F v

⌘i

≤0

⇤ ˜ F



$

1˜ F·n ˆ 2 ⇤ ˜ F

◆T

$

U

1˜ F·n ˆ 2

◆T

1 T + U A U 2 U = aN2

0

0

d

⇣$ ⌘i ˜v · n UT F ˆ

Navier-Stokes Part

Sufficient, but not necessary





U

˜v · n UT F ˆ

$

Re

˜ ⇤,T U + U⇤,T F ˜v · n F ˆ v

Euler Part

⇣$ ⌘ ⇣$ ⌘i ˜v · n ˜v · n F ˆ ⇣ $UT ⌘ F ˆ ⇣$

˜ ⇤,T U + U⇤,T F ˜v · n F ˆ v

U

⇣$ ⌘i ˜v · n UT F ˆ

)

Examples • Euler Inflow/Outlfow • Euler Free-Slip Wall • Navier-Stokes Inflow-Outflow • Navier-Stokes Wall

Linear-Symmetric Equations (

)

! = U ,V ,W ⋅ nˆ ⎡ ⎢ ⎢ U= ⎢ ⎢ ⎢ ⎢ ⎢⎣

ρ U V W P

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ b! ⎢ ⎢ nx ρb + aP ! ⎢ F⋅ nˆ = ⎢ n y ρb + aP ⎢ ⎢ nz ρb + aP ⎢ a! ⎣

( ( (

) ) )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

a2 + b2 = c 2

d t

)

Euler Inflow-Outflow BRIEF ARTICLE THE AUTHOR

Free Stream X

elements

U=0 ||U||2J,N



X Z

2

Boundary faces

@E,N

(

⇤ ˜ F

Free Stream

⌘ 1 ⇣$ ˜ ·n F ˆ 2

U=0 h ⇣$ 1 ˜ ⇤,T ˜v Fv U + U⇤,T F Re

T

U

Specify Free Stream in⇣Upwind Riemann Solver  ⌘ T

) {{

(

F! * UL ,UR = h

(E)

$ 1 ˜ ˜A⋅ ! · nn F F U ˆˆ ! nU ⎡ ⎡ U⎤ ⎤ = A+ UL + A− UR ˆ 2− A⋅ ⎣ ⎦



˜ ⇤,T U + U⇤,T F v ✓

⇤ ˜ F

$

}}

2 ⎣

⇣$ ⌘ ˜v · n F ˆ

1˜ F·n ˆ 2

◆T

U



⇣$ ⌘i ˜v · n UT F ˆ

1 T + U A U 2

0

Euler Free-Slip Wall Specify No Normal Velocity ! = (U ,V ,W )⋅ nˆ = 0

(E)

⎡ 0 ⎢ ⎢ nx bρ + aP ⎢ * F = ⎢ n y bρ + aP ⎢ ⎢ nz bρ + aP ⎢ 0 ⎣

( ( (

⎛ * 1 ⎞ ⎡ U ⎜ F − F⋅ nˆ ⎟ = ⎢ ρ U V 2 ⎝ ⎠ ⎣ T

) ) )

W

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ −b! ⎢ ⎢ nx bρ + aP ⎢ ⎤ P ⎥ ⎢ n y bρ + aP ⎦⎢ ⎢ nz bρ + aP ⎢ −a! ⎣

( ( (

) ) )

⎤ ⎥ ⎥ ⎥ ⎥=0 ⎥ ⎥ ⎥ ⎦

THE AUTHOR

X

Euler Free-Slip Wall ||U||2J,N



X Z

2

Equal & Opposite in Upwind Riemann Solver

ements

Boundary faces

(

h

@E,N



(

) {{

(E)



⇤ ˜ F



⇤ ˜ F

⇣$ ⌘ ˜ v!· n F A⋅ nˆˆ

}}

1˜ F·n ˆ 2

◆T

$

T

T

U

( v ⋅ nˆ )

ext

⇣$ 1 h ˜ ⇤,T ˜ Fv U + U⇤,T F Re

≡!

ext

U ⇣$ ⌘i ˜v · n UT F ˆ

⎡ ⎡ U⎤ ⎤ = A+ UL + A− UR 2 ⎣⎣ ⎦⎦

! nU ˆ A⋅ − $

⇤ ˜ F

⇤ ˜ F

⌘ 1 ⇣$ ˜ ·n F ˆ 2

˜ ⇤,T U + U⇤,T F v

F! * UL ,UR =

⌘ 1 ⇣$ ˜ ·n F ˆ 2

U

1˜ F·n ˆ 2

◆T

1 T + U A U 2 U=c aN2

0

0

( )

= − v ⋅ nˆ

int

≡ −! int

THE AUTHOR

Navier-Stokes Inflow BRIEF ARTICLE

X

elements

d dt

X

||U||2J,N 

elements

2

Free Stream

X Z

Boundary faces

U=0

2 ||U||J,N

@E,N



⇤ ˜ F

X Z

 h2

(

⇣ ⌘ $ 1 ⇤ ˜ ˜ ·n F F ˆ THE AUTHOR 2

⌘ 1 ⇣$ ˜ ·n F ˆ 2(



⇤ ˜ F⌘

⇣$ ⇤,T ˜ ˜ ⇤,T F U + U ˆ vBoundary @E,NFv · n

(E)



(D)

h

⇤ ˜ F

U

⇣$ 1 h ˜ ⇤,T ˜v · Fv U + U⇤,T F Re

T

U T ⇣ ⌘ $ 1 ˜ ⇣ F$· n ˆ ⌘iU 2T F ˜v · n U ˆ

Specify Ext State

faces

T

h 1 ˜ ⇤,T Fv U + U⇤, Re

 ◆ T ⇣ ⌘ $ T 1 1˜ ⇤ 1$ ˜ ˜ ·n · Tn ˆA+ UU 0 F ˆF U2 FU 2 2

˜ ⇤,T U + U⇤,T F v

⇣$ ⌘ ˜v · n F ˆ

⇣$ ⌘i ˜v · n UT F ˆ

( Z T ⇣ ⌘ h ⇣ $ $ X X 1 1 ⇤ 2( ⇤,T ⇤,T ) ˜ ˜ ˜ ˜ ||U||  2 F F · n ˆ U F U + U F  J,N BRIEF ARTICLE⇣ $ Re⌘ v T ⇣ ⌘ h ⇣$ ⌘i $ X Z 2 1 1 @E,N ⇤ Boundary ⇤,T ⇤,T ˜ T ˜ ments ˜ ˜ ˜ F F · n ˆ U F U + U F · n ˆ U Fv · n ˆ faces v v 2 Re @E,N ndary  THE AUTHOR T ⇣ ⌘ $ ces 1 ˜ ⇤ ˜ F F·n ˆ U  T ⇣ ⌘ $ 2 1 ⇤ ˜ ˜ ·n F F ˆ U 2 h ⇣ $ (⌘ ⇣$ ⌘i ⇤,T ⇤,TZ ˜ T ⇣$ T ⌘n h ˜ ˜ X X F U + U F · n ˆ U F · ˆ d 1 1 ⇤ v v (D) =0 2 ⌘ v ˜ ˜ ˜ ⇤,T U + U⇤, ⇣ ⇣ ⌘i $ $ ||U||J,N  2 F F·n ˆ U F v ⇤,T dt ⇤,T ˜ T ˜ 2 Re U +elements U Fv · n ˆ U Boundary Fv · n ˆ @E,N

Navier-Stokes Inflow

v



faces



$

˜⇤ = F ˜v · n F ˆ v

˜ F

Compute Flux ◆T $ ⇣ ⌘ 1 ˜from Interior 1 T h +⇤,T ˜U U + ˜v · n F·n ˆ U U AF 0 U⇤,T F ˆ $





⇣ $ *⌘ T 1 ⇤ = 0U ˜ ˜U F F ·n ˆ 2 Specify Solution

2

(1)⇤

˜ F

2

◆T

v



⇤ ˜ 1$ (E) + (D) = 2 F ˜ F·n ˆ U = aN 0 2

$

1˜ F·n ˆ 2

◆T

U

⇣$ ⌘i ˜v · n UT F ˆ 1 T + U A U 2

0

d dt

X

elements

X Z THE AUTHOR

||U||2J,N 

BRIEF ARTICLE (

2



⇣$ ⌘ 1 ⇤ ˜ ˜ ·n F F ˆ 2 THE AUTHOR

T

U

⇣$ 1 h ˜ ⇤,T ˜v Fv U + U⇤,T F Re

Navier-Stokes Outflow (

X Z

2 X @E,N2 d Boundary faces ||U||J,N dt  elements ⇤ ˜ F

Boundary faces

˜ F



@E,N



⇣$ ⌘ T 1 ˜ ˜ ·n ⇣ ⇣$ ⌘ F ˆ1 h U $ F⌘ T 1 ˜ ˜ ⇤,T U + U⇤,T Free ˜v · n F·n ˆ 2U F F ˆ v ( ⇤



UT F

 Re ⇣ $ ⌘ T h Stream 1 1 h ⇣ $ ˜⌘⇤ ⇣˜$ ⌘i ⇤,T ⇤ ˜  F 2 F F · n ˆ U F U + U T ˜ v ˜ ⇤,T U + U⇤,T F ˜v · n ˆ U F · n ˆ U = 0 v 2 Re v T @E,N ⇣ ⌘ $Boundary 1 ˜ faces F ·n ˆ U ◆ ✓ T $ 2 T ⇣ ⌘ $ 1 1 ⇤ 1 ˜UT A+ U 0 ⇤ ˜ ˜ ˜ ) F F · n ˆ U (E) F F ·n ˆ U 2 2 h ⇣$ ⌘ ⇣ $ 2⌘i T ˜ ˜ ⇤,T U + U⇤,T F ˜v · n F ˆ U Fv · n ˆ v h ⇣$ ⌘ ⇣$ ⌘i T ˜ ˜ ⇤,T U + U⇤,T F ˜v · n F ˆ U Fv · n ˆ $ v (D) =0 ⇤ ˜ =F ˜v · n F ˆ v

2X Z

˜⇤ = 0 F v ✓



1

Specify Flux ◆

$

T

U =U *

1

Use interior Solution T

+

˜ ⇤ =˜ F ˜ ·n F ⇤,T ⇤,T ˜ Fv U + U v Fv ·v n ˆˆ

˜v · n UT F ˆ

˜ ⇤ =$ F 0 ⇤v

(1)

˜ =F ˜v · n F ˆ v

Navier-Stokes Wall ✓

$

◆T

1˜ ˜U⇤ = 10UT A+ U F·n ˆ F v 2 2

˜⇤ F

0

◆T◆T ✓ ✓ $1 $ 1 2T + ⇤ ˜⇤1 ˜ ˜ ˜ F F · n ˆ U = aN F F2· n ˆ U U A 0U 2 2

Satisfy (E) through Riemann Solver0

(1) (2)

Re-Write (D) (3) (2) ˜ ⇤ + U⇤,T UT F v

(4)

(3)



˜ F



⇣$ ⌘ ˜⇢v · n F ˆ ⇤ ˜ F

⇣$ ⌘ ˜v · n F ˆ

⌘o ✓1 ⇣ $ ⌘$T ◆T 1 n T ⇣ $ ˜ ·n ˜v · n ⇤F ˆ1 ˜ U U 2F ˆ ˜ F·n ˆ Re U = aN 0 2F 2 ⇣$ ⌘ n T ˜ T ˜⇤ U F · n ˆ = U Fnv v T ⇣$ ⌘

1 ˜ F·n ˆ 2

˜ ⇤ + U⇤,T UT F v

(5) ⇥

⇢ U

V

W

2

6 ⇤6 P 6 6 4

0 ⌧1⇤ ⌧2⇤ ⌧3⇤

r

@P ⇤ @n

⌧1 ⌧2 ⌧3

@P @n

3

⇤,T ˜v · n F ˆ + U ⇣$ ⌘o

1 ˜v · n UT F ˆ Re

⇣$ ⌘ n ˜v · n ˜⇤ UT F ˆ = UT F v

Written out…

(4)

U

o

$

$

o

⇣$ ⌘ ˜v · n F ˆ 0

˜v · n F ˆ + U⇤,T

⇣$ ⌘ ˜v · n F ˆ 0

2

71 6 7 ⇥ ⇤ ⇤6 7 + ⇢ U⇤ V ⇤ W⇤ P⇤ 6 7 6 5 4 1

0 ⌧1 ⌧2 ⌧3 r @P @n

3

7 7 7 6 0, 7 5

τ i = nxτ i1 + n yτ i2 + nzτ i3



(3)

˜ F

⌘ 1 ⇣$ ˜ ·n F ˆ 2



⇣$ ⌘ ˜v · n F ˆ

T

U

⌘o 1 n T ⇣$ ˜v · n U F ˆ Re

⇣$ ⌘ n ˜v · n ˜⇤ UT F ˆ = UT F v

o

⇣$ ⌘ ˜v · n F ˆ 0

Navier-Stokes Wall

˜ ⇤ + U⇤,T UT F v

(4) (5) ⇥

⇢ U

V

W

2

6 ⇤6 P 6 6 4

0 ⌧1⇤ ⌧2⇤ ⌧3⇤

r

@P ⇤ @n

⌧1 ⌧2 ⌧3

@P @n

$

˜v · n F ˆ + U⇤,T

3

2

7 6 7 ⇥ ⇤ 6 ⇤ ⇤ ⇤ ⇤ ⇤ 7+ ⇢ U V 6 W P 7 6 5 4 1

Vanishes if τ i* = τ i

U* = V * = W * = 0

+ Isothermal

Adiabatic

P* = 0

P* = P

∂P * ∂P = ∂n ∂n

∂P * =0 ∂n

0 ⌧1 ⌧2 ⌧3 r @P @n

3

7 7 7 6 0, 7 5

d dt

X

elements

2 ||U||J,N





2

T ⇣ ⌘ $ 1 ⇤ ˜ ˜ ·n F F ˆ (U Z X2 ⇤

˜ F

⇣ ⌘ $ 1 ˜ F·n ˆ 2 ⌘i

Not so fast… h



Boundary $ ⇤,T faces

˜ ⇤,T U + U F v

@E,N ⌘

˜v · n F ˆ



⇤ ˜ ˜⇤ = F ˜v F F ·n ˆ v $

⇣$ ˜v · n UT F ˆ

⇣ ⌘ $ 1 ˜ F·n ˆ 2

h 1 ˜ ⇤,T Fv Re

T

U

T

U

A Guide to the Implementation of Boundary Conditions… AIAA 2014

Use interior ✓

(1)

(2)

Viscous Flux

(3)

(2)

(SSC)



⇤ ˜ F

$

◆T

1˜ 1 T + $ F·n ˆ U U˜ ⇤A U˜ 0 2 from Interior:2 Fv = Fv · nˆ



(1)

˜⇤ = 0 F v

⇣$ ⌘ ˜ ⇤,T U + U⇤,T *F ˜v · n F ˆ v U =U Solution

⇤ ˜ F

h

⇤ ˜ F

1✓˜ F ·⇤n ˆ ˜ 2 F $

◆T

◆2T

$

0 1U˜= aN F·n ˆ U 2

⇣$ ⌘i ˜v · n UT F ˆ

1 T + U A U 2

??? T ⇣ ⌘ n ⇣ ⌘o $ $ 1 ˜ 1 T ◆˜ ✓ F·n ˆ U U FTv · n ˆ $ 1˜ ⇤ Re 2 ˜ F F·n ˆ U = aN2 0

≥0

2

≥0

???

0

Nonlinear: Entropy Bound 2

THE AUTHOR

d S6 dt (6)

2

Boundary faces

1 + Re

d (7) n dt S 6 ⇣ T (W)

(6)

X Z n



˜ ec,⇤ (W)T F

N

⇣ $ ⌘⌘ ⇣ ! ⌘o ˜ ·n F ˆ + F˜ ✏ · n ˆ dS

⇣$ ⌘⌘ ⇣$ ⌘o X Z n THE⇣AUTHOR ˜⇤ ˜v · n ˜v · n WT F F ˆ + (W⇤ )T F ˆ dS , v

Boundary faces

X Z n

N



⇣ $ ⌘⌘ ⇣ ! ⌘o ˜ ·n ˜✏ ·n F ˆ + F n ⇣ ⇣ $ ˆ ⌘⌘ dS

T ˜ ec,⇤ (W) F ⇣ $ ⌘⌘ ⇣ ! ⌘o 1 ⇣$ ⌘o T ˜⇤ ⇤ T ˜ ec,⇤ Boundary ˜ ·n ˜v · n ˜v · n ˜✏ ·n F F ˆ + F ˆ + W F F ˆ + (W ) F ˆ 0 N v faces Re Z n ⇣ ⇣$ ⌘⌘ ⇣$ ⌘o X 1 T ⇤ ˜⇤ ˜v · n ˜v · n + WT F F ˆ + (W ) F ˆ dS , v Re Boundary

Sufficient Condition for Stability faces

(7)

n



˜ ec,⇤ (W)T F

N

⇣ $ ⌘⌘ ⇣ ! ⌘o n ⇣ 1 ˜ ·n ˜⇤ F ˆ + F˜ ✏ · n ˆ + WT F v Re

⇣$ ⌘⌘ ⇣$ ⌘o ˜v · n ˜v · n F ˆ + (W⇤ )T F ˆ 0

TO DO • Robin Conditions • Entropy BCs Issues: Linear theory well understood Linear Stability

⇔ Nonlinear Stability

Entropy function not unique A stable procedure with one entropy function may not be stable with another.

Summary Have linear and entropy stability bounds to establish stability of DGSEM Approximations for • Arbitrary 3D geometries • Curved elements • Any polynomial order Bounds establish stable boundary procedures Approach extends to, Shallow water, MHD eqns., … See Andrew’s talk