Towards an Entropy Stable Spectral Element ...

Overview/Motivation Summation-by-parts Operators Entropy Stability Staggered Operators Nonconforming Interfaces

Towards an Entropy Stable Spectral Element Framework for CFD Mark H. Carpenter, Comp. AeroSci. Branch, (CASB), NASA LaRC, Hampton, VA, U.S.A. Matteo Parsani King Abdullah Univ. of Science and Technology (KAUST) Travis C. Fisher, Sandia National Laboratories, Albuquerque, NM Eric J. Nielsen, CASB

SciTech-2016, San Diego, CA, U.S.A., 01/06/2016 Carpenter, Mark

Entropy Stable SBP


Overview/Motivation Summation-by-parts Operators Properties Navier-Stokes Eqns. Polynomial SBP Operators Entropy Stability Continuous Entropy Semi-Discrete Entropy High-Order Generalization Staggered Operators Multiple Dimension Test Problems: Staggered vs. Conventional Nonconforming Interfaces

Carpenter, Mark

Entropy Stable SBP


Overview/Motivation

Carpenter, Mark

Entropy Stable SBP


Overview/Motivation I

High-Order spatial operators I I I

Well suited for time-dependent simulations with scale separation Informal poll: Fragile in real world applications Incomplete nonlinear stability theory to guide development I I I

I

I

Use without stabilization: Compromised Robustness Adopt low-order stabilization techniques (Compromised Accuracy) Filter, de-aliasing, over-integration, spectral hyper-viscosity . . .

Move from experimentation to mathematics!

Entropy conservation/stability theory (2013 Fisher, WENO FD) I I

Two decades of development for low-order methods (e.g., Tadmor) Summation-by-parts - SAT operators: All Diagonal Norm operators I I I I

I

Nonlinear Stability of Navier-Stokes equations Arbitrary order of accuracy Complex Geometries via mapped hexahedral multi-domains Strong Conservation Form (amenable to shock capturing)

Comparison approach (e.g., WENO (+) entropy stable operators)

Carpenter, Mark

Entropy Stable SBP


Progress Toward an SS Framework: Past, Present, Future I

Past accomplishments I

Entropy stable spectral elements: recent work (2013, 2014) I I I I

I

Current tasks I

Staggered grid operators I I

I

Broader selection of collocation points (Legendre-Gauss vs. LGL) Data movement in element while maintaining Entropy stability

Nonconforming interfaces I

I

Curvilinear mapped hexa elements (Legendre-Gauss-Lobatto) Strong Conservation Form (amenable to shock capturing) Comparison approach (e.g., Nodal DG and entropy stable operators) Solid Wall Boundary Conditions that preserve SS estimate

p-refinement ; h-refinement ; sliding

Future tasks I

Multi-dimensional SBP operators I

I

Triangle, (Prism), Tetrahedra, Cut cell, Overset grids

Turbulence modeling Carpenter, Mark

Entropy Stable SBP


Properties Navier-Stokes Eqns. Polynomial SBP Operators

Summation-by-parts Operators

Carpenter, Mark

Entropy Stable SBP



Mimetic Operators SBP derivative operators discretely mimic the integration-by-parts Z xR Z xR xR φx q dx φqx dx = φq|xL − xL

xL

Mimetic form for first derivative Dφ is D = P −1 Q, T

P = PT ,

Q = B − Q,

ζ T Pζ > 0,

ζ 6= 0,

B = diag (−1, 0, . . . , 0, 1)

Semi-Discrete: φT PP −1 Qq = φT B − QT q = φN qN − φ1 q1 − φT DT Pq

Lemma

All SBP derivatives are discretely conservative in the P-norm. Carpenter, Mark

Entropy Stable SBP



Telescoping Flux Form fx (q) = P −1 Qf + Tp+1 = P −1 ∆f + Tp+1 N × (N + 1) matrix ∆ is defined as  −1 1 0  0 −1 1   .. ∆= 0 . 0   0 0 0 0 0 0

0 0 .. . −1 0

0 0 0 1 −1

Calculates undivided difference of adjacent fluxes

 0 0    0   0  1

Lemma All (tensor product) SBP differentiation matrices are telescoping operators in the norm P and may be expressed as above. Carpenter, Mark

Entropy Stable SBP



Calorically perfect Navier-Stokes Equations qt + (f i )xi = (f (v )i )xi , Bq = gb ,

x ∈ Ω,

x ∈ ∂Ω,

q(x, 0) = g0 (x),

t ∈ [0, ∞),

t ∈ [0, ∞), x ∈ Ω,

SBP operators and equivalent telescoping form for N-S yields i (v )i qt = −Di [fi (q)] + Di [c]ij Dj q + P −1 gb = P −1 ∆i −f + f + P −1 gb q(x, 0) = g0 (x),

x ∈Ω

with gb containing BC data

Carpenter, Mark

Entropy Stable SBP



Example: LGL P4 Differentiation D = P −1 Q , P = δx diag 

  Q=  

− 7 120

7 120

−1 2 √ 7 + 21

4 15 √ −7 + 21 1 20

7 120

1 10

√ 7 + 21

0 q 28 7 3 − 45 q 49 7 3 180 √ 7 −7 + 21 120

Carpenter, Mark

49 90

− 4 15 q 28 7 3 45 0 q 28 7 3 − 45 4 15

32 45

49 90

1 10

√ −7 + 21 q 7 49 3 − 180 q 28 7 3 45

− 7 120

− 7 120

0 √ 7 + 21

Entropy Stable SBP

− 7 120

7 120

− 1 20 √ −7 + 21 − 4 15 √ 7 + 21 1 2

     



Legendre Spectral Collocation: LGL P4 Differentiation f¯2

f¯0 f¯1 f1 u1

−1

Lj (x) =

P =

Z

x3

x4 x ¯3

x ¯2

−9 10

−

YM

p3 7

x − xek − xek

k =1 e k 6=j xj

1

f4 u4

f3 u3

x2

x1 x ¯0 x ¯1

−16 45

+16 45

0

7

x5 ¯5 x ¯4 x +9 10

+1

>

D = P −1 Q , >

;

Q =

−1

P =

+

p3

L(ηl ; x) = (L1 (ηl ; x), . . . , LN (ηl ; x))

;

L(x; x) L(x; x) dx

X

f¯4 f¯5 f5 u5

f¯3

f2 u2

Z

1

>

L(x; x) L(x; x)0 dx

−1

>

L(ηl ; x)L(ηl ; x) ω`

`

;

Q =

X `

Carpenter, Mark

Entropy Stable SBP

0

>

L(ηl ; x)L (ηl ; x) ω`



Legendre Spectral Collocation: Interpolation On interval −1 ≤ x ≤ 1, define discrete points ˜ = xe1 , xe2 , · · · , xeM−1 , xeM x

> >

, −1 ≤ xe1 , xe2 , · · · , xeM−1 , xeM ≤ 1;

x = (x1 , x2 , · · · , xN−1 , xN ) , −1 ≤ x1 , x2 , · · · , xN−1 , xN ≤ 1.

“Magic” interpolation operators exchanging data ILGL2G = P˜ −1 RLG−LGL , IG2LGL = P −1 R> LG−LGL , > P˜ ILGL2G = ILG→LGL P,

where P˜LG =

R1

−1

R1 > ˜) L(x; x ˜)> dx ; PLGL = −1 L(x; x L(x; x) L(x; x) dx R1 > ˜) L(x; x) dx. RLG−LGL = −1 L(x; x Carpenter, Mark

Entropy Stable SBP


Continuous Entropy Semi-Discrete Entropy High-Order Generalization

Entropy Stability

Carpenter, Mark

Entropy Stable SBP



The Stability of High-Order Schemes: Navier-Stokes I

Mathematical Entropy (Continuous: Discrete) I I I

I

Convex extension of original equations (Friedrichs / Lax) Formed by contracting N-S Eqns. with entropy variables Bounded physical quantity. (N-S Eqns. -> thermodynamic entropy)

What does it buy you? I

L2 Stability I I

I I

I

Entropy is bounded from above in an integral sense Entropy Convexity → integral bounds on conserved variables

Neutral Nonlinear Stability of Euler Eqns! The code doesn’t “blow up” (well, usually. . . see below)

What it doesn’t guarantee I I I

Stable boundary conditions (Svärd, Parsani) No bounds on derivatives; (e.g., KE stability) Positivity (negative temperatures: Shu’s limiters)

Carpenter, Mark

Entropy Stable SBP



The Continuous Entropy Equation S(q) is convex and when differentiated, simultaneously contracts all (inviscid and viscous) spatial fluxes as follows Sq fxi i = Sq fqi qxi = Fqi qxi = Fxi i

;

i = 1, · · · , d

entropy variables : w T = Sq Contracted equation is differential form of entropy equation (v ) Sq qt + Sq f (q)xi = St + Fxi = Sq fxi = w T f (v ) − wxTi cîj wxi xi

Global integration: conservation of entropy Z Z h i d S dxi = w T f (v ) − F − wxTj cîj wxi dxi dt Ω ∂Ω Ω The matrix cˆ is symmetric positive definite (S.P.D.) and dissipative Carpenter, Mark

Entropy Stable SBP



Semi-Discrete Analysis: Inviscid Fluxes wT Pqt + wT ∆i fi = wT ∆i f

(v )i

+ wT g b ,

i = 1, 3

Inviscid Fluxes wT ∆f = F (qN ) − F (q1 ) = 1T ∆F

Theorem The local conditions, (e.g., Tadmor for 2nd-order) T (S)

(wi+1 − wi ) fi

= ψ˜i+1 − ψ˜i ,

i = 1, 2, . . . , N −1; ψ˜1 = ψ1 ,

when summed, telescope across domain conserving entropy

Carpenter, Mark

Entropy Stable SBP

ψÑ = ψN



Semi-Discrete Analysis: Generalized Inviscid Fluxes Theorem A two-point high-order entropy conservative flux satisfying T (S) (wi+1 − wi ) ¯fi = ψ˜i+1 − ψ˜i may be constructed as ¯f (S) = i

N X i X

2q(`,k )¯fS (q` , qk ) ,

1≤i ≤N −1

k =i+1 `=1

¯fS (q` , qk ): a two-point function that satisfies entropy conservation (w` − wk )T ¯fS (q` , qk ) = ψ` − ψk High-order entropy conservative flux satisfies (S)

wT P −1 ∆f

= P −1 ∆F = Fx (q) + Td

Design order for all variables Carpenter, Mark

Entropy Stable SBP


Multiple Dimension Test Problems: Staggered vs. Conventional

Staggered Operators

Carpenter, Mark

Entropy Stable SBP



History of Staggered Grid Operators f¯2

f¯0 f¯1 f1 u1

f2 u2

I

I

u ˜2

u ˜3

u ˜4

x ˜1

x ˜2

x ˜3

x ˜4

−9 10

x3

x2 −

p3 7

x4

x ¯2

x ¯3

−16 45

+16 45

0

+

p3 7

x5 ¯5 x ¯4 x +9 10

+1

Bernardi/Maday (1988), Maday/Patera/Ronquist (1992) Fischer (1994) (p:p+2 basis)

Extension into compressible equations I I

I

f¯4 f¯5 f5 u5

Popular design philosophy: stokes & incompressible literature I

I

f4 u4

u ˜1

x1 x ¯0 x ¯1 −1

f¯3 f3 u3

Cai/Gottlieb/Harten (1992) Chebyshev Volume scheme Kopriva/Kolias (1996) Chebyshev points enhanced robustness

Evolution/maturation for compressible eqns I I

Spectral differencing: Liu/Vinokur/Wang (2006), May/Jameson Flux Reconstruction: Huynh (2007), Vincent/Castonguay/Jameson (2011) Carpenter, Mark

Entropy Stable SBP



Why Staggered I I

Higher accuracy on Legendre-Gauss points than LGL Greater Flexibility I

I I

Potentially no vertex points in operator (corner singularities)

More Robust (????) Nonconforming interfaces I

Desirable structural properties allow trivial extension of SBP ideas ILGL2G = P˜ −1 RLG−LGL , IG2LGL = P −1 R> LG−LGL ,

> P˜ ILGL2G = ILG→LGL P,

Carpenter, Mark

Entropy Stable SBP



A Staggered Grid Algorithm: (Kopriva) f¯2

f¯0 f¯1 f1 u1

f2 u2

f4 u4

f¯4 f¯5 f5 u5

u ˜1

u ˜2

u ˜3

u ˜4

x ˜1

x ˜2

x ˜3

x ˜4

−9 10

x3

x2

x1 x ¯0 x ¯1 −1

f¯3 f3 u3

−

p3 7

−16 45

x4 x ¯3

x ¯2 0

+16 45

+

p3 7

x5 ¯5 x ¯4 x +9 10

+1

˜ (solution) and x (fluxes). Define two sets of collocation points: x I Interpolate the discrete entropy variables from x ˜ to x. I

Build the nonlinear fluxes f and f (V ) on the set of points x.

I

Build interface and/or boundary penalties at boundaries of x.

I I

Differentiate fluxes on x; impose penalties using SAT approach. ˜. Interpolate discrete flux derivatives and penalties back to x

I

˜. Advance solution in time using interpolated flux derivative on x Carpenter, Mark

Entropy Stable SBP



Stability of Staggered Grid Algorithm: Burgers’ Eqn. u t +f (u)x = [εf (v ) ]x ,

f (u) =

u2 , 2

f (v ) = ux ,

x ∈ [xL , xR ],

t ∈ [0, ∞),

Energy ˜ (u = IG2LGL u ˜t + u

1 3

;

ILGL2G (D [u] + [u] D) u

Entropy d ˜ + dt u

[u] = Diag[u]) = ILGL2G D [] D u − ILGL2G (PenO ) P −1 e(xL ) + ILGL2G (PenN ) P −1 e(xR )

ILGL2G P −1 ∆ f(u) = ILGL2G D [] D u + PenO + PenN

˜ LGL2G = IG2LGL > P Stability: PI 1 > d ˜> ˜ ˜ dt u P u + 3 u (Q [u] + [u] Q) u Carpenter, Mark

=

Entropy Stable SBP

−u> D> [] D u



Staggered Grid Operators: Extension to 2D (-1,+1)

(-1,-1)

(+1,+1)

(-1,+1)

(+1,-1)

(-1,-1)

(a) Fully-staggered.

(+1,+1)

(+1,-1)

(b) Semi-staggered.

Figure : Fully- and semi-staggered 2D tensor product elements.

Both used herein. Semi-staggered extended to nonconforming Carpenter, Mark

Entropy Stable SBP



Staggered Grid Algorithm: Navier-Stokes Eqn. Semi-Discrete NS on two elements. LDG Viscous formulation h i ˜` dq (Int),q + ILGL2G ` Px−1 ∆xi ,` fi,` − Dxi ,` [b cij,` ] Θj,` = ILGL2G ` Px−1 gi,` , i ,` i ,` dt (Int),Θ

Θi,` − Dxi wl = Px−1 gi,` i ,`

,

i h ˜r dq (Int),q b + ILGL2G r Px−1 ∆ f − D [ c ] Θ = ILGL2G r Px−1 gi,r , x ,r x ,r i,r ij,r j,r ,r i i i i ,r dt (Int),Θ

Θi,r − Dxi wl = Px−1 gi,r i ,r

,

Nonlinear Stability Estimate

q

2 q

2

d e> b b (I) (V ) ˜ l +1 ˜ r )+2 ( [b e> P

˜l S ˜r S b (1 P c ]Θ + [ c ]Θ , ij,l j,l ij,r j,r ) = Υ +Υ

dt Pl Pr

where Υ(I) and Υ(V ) are the inviscid and the viscous interface terms. Carpenter, Mark

Entropy Stable SBP



Staggered Grid Algorithm: LDG/IP Penalty Terms (Int),q

g1,l

(Int),q

g1,r

h (−) i 1 h (−) i (−) h (+) i (+) (−) (−) (+) b = +f1 − fssr (qi , qi ) e(−) − c1,j Θj − b c1,j Θj e 2 1 + [L] w(−) − w(+) e(−) , 2 1 (−) (Int),Θ (+) g1,l = − w −w e(−) , 2 h (+) i 1 h (+) i (+) h (−) i (−) (+) (−) (+) b = −f1 + fssr (qi , qi ) e(+) + c1,j Θj − b c1,j Θj e 2 1 (+) [L] w − w(−) e(+) , + 2 1 (+) (Int),Θ g1,r = + w − w(−) e(+) . 2 Carpenter, Mark

Entropy Stable SBP



Test Problems I

Isentropic Euler vortex propagation (Uniform / Nonuniform) I

I

I

I

Exact Solution: Testing Accuracy of conventional and staggered

Taylor Green Vortex (Uniform / Nonuniform) I

I

Exact Solution: Testing Accuracy of conventional and staggered

Viscous Shock propagation (Uniform / Nonuniform)

Model of Turbulent cascade. Torture testing of algorithms

Blunt Body in supersonic crossflow I

Torture testing of algorithms

Carpenter, Mark

Entropy Stable SBP



Smooth 3D Grids: Euler and NS, Similar Behavior Grid convergence study: Euler vortex on Cartesian uniform grids

Fully-staggered, pLG =1, pLGL =2 Fully-staggered, pLG =2, pLGL =3 Fully-staggered, pLG =3, pLGL =4 Fully-staggered, pLG =4, pLGL =5 Fully-staggered, pLG =5, pLGL =6 Fully-staggered, pLG =10, pLGL =11 Conventional, pLGL =1 Conventional, pLGL =2 Conventional, pLGL =3 Conventional, pLGL =4 Conventional, pLGL =5 Conventional, pLGL =10

Error (L2 -norm)

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12

10-2

1/(DOFs)1/3

10-1

100

(a) Euler Vortex


Error (L2 -norm)

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

Grid convergence study: Viscous shock on Cartesian uniform grids

10-2

1/(DOFs)1/3

10-1

(b) Viscous Shock Carpenter, Mark

Entropy Stable SBP

100



Conventional/Staggered: Viscous Shock, Smooth Grid Grid convergence study: Viscous shock on Cartesian uniform grids


Error (L2 -norm)

100 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

10-2

1/(DOFs)1/3

Carpenter, Mark

10-1

Entropy Stable SBP

100



Discontinuous 3D Grids: Conventional VS. Staggered Grid convergence study: Euler vortex on unstructured nonuniform grids


Error (L2 -norm)

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

10-1

100

1/(DOFs)1/3

(c) Euler Vortex


Error (L2 -norm)

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

Grid convergence study: Viscous shock on unstructured nonuniform grids

10-1

10-2

1/(DOFs)1/3

(d) Viscous Shock Carpenter, Mark

Entropy Stable SBP

100



Conventional/Staggered: Viscous Shock, Random Grid Grid convergence study: Viscous shock on unstructured nonuniform grids


Error (L2 -norm)

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 10-12 10-13

10-2

10-1

1/(DOFs)1/3

Carpenter, Mark

Entropy Stable SBP

100



Taylor Green 3D Vortex: Staggered Vs. Conventional Initial condition: x x x 1 2 3 u1 = V0 sin cos cos , L L L x x x 1 2 3 u2 = −V0 cos sin cos , u3 = 0, L L L ρ0 V02 2x1 2x2 2x3 p = p0 + cos + cos cos +2 16 L L L Pathological Grid: (Torture)

Carpenter, Mark

Entropy Stable SBP

(3)



Taylor Green 3D Vortex: Staggered/Conventional (cont) Taylor-Green vortex, Re =800, M =0.08

0.01

0.008

0.008

dke/dt

dke/dt

Fully-staggered, pLG =15, pLGL =16, 42 cells DNS, Carton de Wiart et al.

0.012

0.01

0.006

Fully-staggered, pLG =15, pLGL =16, grid: 43 Fully-staggered, pLG =16, pLGL =17, grid: 43 DNS, Brachet et al.

0.004 0.002 0.0

Taylor-Green vortex, Re =1,600, M =0.08

0.014

0.012

0.006 0.004 0.002 0.0

0

2

4

6

Time

8

10

(e) Re=800

0

5

10

Time

15

(f) Re=1,600

Carpenter, Mark

Entropy Stable SBP

20



Cylinder in Supersonic Cross-flow

Mach 1 0

Density 2

0.4

1.2

1.6

2

0.0561

2.42

(g) Mach number; ∆M = 0.0095.

0.8

2.36

(h) Density; ∆ρ = 0.0090.

Figure : Flow past 3D square cylinder at Re∞ = 104 and M∞ = 1.5; fourth-order accurate fully-staggered method without stabilization.

Staggered and conventional: equivalent robustness Carpenter, Mark

Entropy Stable SBP


Nonconforming Interfaces

Carpenter, Mark

Entropy Stable SBP


Nonconforming Interfaces

Figure : Non-Conforming semi-staggered 2D collocation and flux points.

Carpenter, Mark

Entropy Stable SBP


Re-tooling SBP-SAT Entropy Stability N

Consider 2D set of points Xxy = [(xi , yi )]i=1 , and monomial basis Smn = x m y n ,

0≤m+n ≤p

and their projection onto the grid Xxy as follows: > Smn (Xxy ) = (smn (x1 , y1 ), . . . , smn (xN , yN )) , > 0 0 0 Smn (Xxy ) = (smn (x1 , y1 ), . . . , smn (xN , yN )) .

Definition Dx is an SBP approx of I

Dx = P

∂ ∂x

at Xxy if

Qx , Qx + Qx = Bx , P = P T , ζ > Pζ > 0 , H and Smn Tk Bx Smn ` = Γ Smn k Smn ` ~nx dΓ ,

−1

T

I

Bx = Bx

I

0 Dx Smn = Smn , 0 ≤ m + n ≤ p or equivalently PN m−1 n m n yi , 0 ≤ m + n ≤ p . `=1 q`k xk yk = P(i)(i) m xi Carpenter, Mark

Entropy Stable SBP

ζ 6= 0,


Building an SBP operator at nonconforming interface??? ¯ x (with no interface SAT penalty) is given by D  L −1 L L (PLG QLG ) ⊗ ILG 0  −1 H ¯ H H Dx =  0 (PLG QLG ) ⊗ ILG 0 0

Carpenter, Mark

Entropy Stable SBP

 0   0 L −1 L L (PLG QLG ) ⊗ ILG

,


Building an SBP operator at nonconforming interface! SAT Penalty terms at block 2x2 interface L L L PLG 0 −PLG PLG IH2L 1 1 + = H H H 2 2 0 −PLG −PLG IL2H PLG

1 2

0 −R> L−H

Combined operator satisfies the following structural constraints: Dx = Px−1 Qx

;

(a) No SAT

Carpenter, Mark

Qx + Qx T = Bx .

(b) With SAT

Entropy Stable SBP

RL−H 0


Where’s the Flux! Generalizing Entropy Stability Theory Recall high order entropy conservative flux ¯f (S) = i

N X i X

2q(`,k )¯fS (q` , qk ) ,

1 ≤ i ≤ N − 1,

k =i+1 `=1

when the two-point non-dissipative function (from Tadmor) is used ¯fS (qk , q` ) =

Z

1

g (w(qk ) + ξ (w(q` ) − w(qk ))) dξ,

0

Carpenter, Mark

Entropy Stable SBP

g(w(u)) = f (u).


Non tensor-product entropy stability Theorem The multidimensional discrete derivative operator DxS , N

X ∂fi ≈ DxS f |i = Pii−1 2q(i,j) fS (ui , uj ) 1 ≤ i ≤ N, ∂x j=1

achieves the design order accuracy p of SBP operator Dx = P −1 Qx , provided fluxes fS (ui , uj ) are integrations through phase space ξ fS (uk , u` ) =

Z

1

g (w(uk ) + ξ (w(u` ) − w(uk ))) dξ,

g(w(u)) = f (u).

0

The coefficient q(i,k ) corresponds to the (i, j) row and column in the SBP operator Qx , respectively. Carpenter, Mark

Entropy Stable SBP


Non tensor-product entropy stability (cont) Theorem The multidimensional discrete operator DxS given by the expression DxS f = 2P −1 [Qx fs ]1 is entropy conservative provided fluxes fS (u` , uk ) satisfy (wi − wj )¯fS (ui , uj ) = ψi − ψj . Operator satisfies additional local entropy consistency property, 2[W ]P −1 [Qx fS (ui , uj )]1 = 2P −1 [Qx FS ]1 = Fx + O ((δx)p ) , where 2P

−1

[Qx FS ]1 =

Pii−1

N X j=1

2q(i,j)

(ψi + ψj ) (wi + wj ) fS (ui , uj ) − , 1 ≤ i ≤ N. 2 2

Carpenter, Mark

Entropy Stable SBP


Nonconforming Entropy Flux Convergence Study qt + (f i )xi = 0,

x ∈ Ω,

t ∈ [0],

q(x, 0) = g0 (x) = Vortex

Table : Error convergence: semi staggered algorithm for nonconforming interface pL = 4, 5, 6, pH = 5, 6, 7; Euler vortex propagation. Resolution 14 15 16 17 18 19 20

pL = 4, pH L2 error 4.29E-004 3.40E-004 2.73E-004 2.20E-004 1.80E-004 1.48E-004 1.22E-004

=5 L2 rate -3.29 -3.37 -3.44 -3.50 -3.55 -3.60 -3.78

pL = 5, pH L2 error 1.01E-004 6.56E-005 4.65E-005 3.27E-005 2.37E-005 1.74E-005 1.30E-005

Carpenter, Mark

=6 L2 rate -4.13 -6.32 -5.33 -5.82 -5.63 -5.70 -5.68

Entropy Stable SBP

pL = 6, pH L2 error 6.33E-006 4.82E-006 3.44E-006 2.55E-006 1.90E-006 1.43E-006 1.08e-006

=7 L2 rate -5.94 -3.96 -5.23 -4.88 -5.21 -5.22 -5.48


Conclusions I

Staggered grid Entropy stability for Compressible NS I

Broader Selection of points including Legendre-Gauss I I

I

Test cases demonstrate design order accuracy for smooth flows I I

I

I

Requires precise restriction/prolongation interpolation pair Automatically satisfied between LG and LGL points Significantly more accurate on random meshes More costly accurate to implement

Equivalent robustness as original LGL operators

Nonconforming interface I I I I I

multi-dimensional extension of SBP operators Constructed SBP operators at nonconforming interfaces reformulated entropy stability theory for multi-dimensional operators proved design order accuracy more coding is required to verify accuracy

Carpenter, Mark

Entropy Stable SBP


Thank you for your attention.

Carpenter, Mark

Entropy Stable SBP


Entropy Stability: SBP-SAT VS. FEM I

SBP-SAT I I I I

I

Strong conservation form operator Summation of magic dyadic entropy fluxes Stability ensured by entropy flux conservation across domain aliasing does not induce instability

FEM I I

Typically weak form operator Stability ensured by integral exactness I I I I

Integration of rational polynomials over element (p ≤ 3) Inexactness -> aliasing -> instability Additional work on integrals stabilizes formulation Estimate error and added viscosity

Carpenter, Mark

Entropy Stable SBP

Towards an Entropy Stable Spectral Element ...

Towards an Entropy Stable Spectral Element ...

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