towards robust, high order and entropy stable algorithms ... · barcelona, spain july 22, 2014....

Towards robust, high order and entropy stable algorithmsfor the compressible Navier–Stokes equations on

unstructured grids

Matteo Parsani, Mark H. Carpenter, and Eric J. Nielsen

Computational Aeroscience BranchNASA Langley Research Center

WCCM XI/ECFD VIBarcelona, Spain

July 22, 2014

Acknowledgments

Travis C. Fisher (Sandia National Laboratories, USA)

Matteo Parsani 1 / 18

Outline

1 Introduction

2 Continuous entropy analysis

3 Summation-by-parts Operators

4 Semi-discrete system

5 Implementation / Applications

6 Conclusions


Introduction

...high-order accurate, nonlinearly stable algorithms for PDEs.... why?

Jet plumes

Photo courtesy of NASA Langley Research Center

Non-petroleum-based fuels

Photo courtesy of SANDIA National Lab


Introduction

...a viable path: High order entropy stable discretizations

Well suited for time-dependent simulations

Well suited for emerging HPC architectures

Little stability theory to guide development (linear or nonlinear)

⇓ Two decades of development for low order methods

Recent Developments

High order accurate entropy stable discontinuous FE of any order p

Strong conservation form for shock capturing

Entropy stable boundary conditions


Introduction

The entropy stability of high order schemes: Navier–Stokes

Mathematical entropy (s)I Convex extension of original equations (Friedrichs / Lax)I Formed by contracting N–S equations with entropy variablesI Bounded physical quantity (thermodynamic entropy)

What does it buy you?I L2 Stability

? Entropy is bounded from above in an integral sense (stability plateau)? The code doesn’t “blow up”

What it doesn’t guaranteeI Stable boundary conditions (Svard and herein)I Positivity (negative temperature and density: Shu’s limiters)


Continuous entropy analysis

Compressible Navier–Stokes equations

qt +(f

(I )i

)xi

=(f

(V )i

)xi, x ∈ Ω, t ∈ [0,∞),

q|∂Ω = g (B)(x , t), x ∈ ∂Ω, q(x , 0) = g (0)(x), x ∈ Ω,

Entropy variables: Sq with S = −ρsEntropy fluxes: Fi = −ρui s

Entropy equation:

Sqqt + Sq(f

(I )i

)xi

= St + (Fi )xi = Sq(f

(V )i

)xi

=(w>f

(V )i

)xi− w>xi cij wxj

Global conservation statement

d

dt

∫

ΩS dx =

[w>f

(V )i − Fi

]∂Ω−∫

Ωw>xi cij wxj dx


Summation-by-parts Operators

Summation-by-parts mimetic operators

Integration-by-parts

∫ xR

xL

φqx dx = φq|xRxL −∫ xR

xL

φxq dx

Semi-Discrete:

φ>PP−1Qq = φ>(B −Q>

)q = φNqN − φ1q1 − φ>D>Pq

Mimetic form for first derivative Dφ if

D = P−1 Q, P = P>, ζ>Pζ > 0, ζ 6= 0,

Q> = B −Q, B = diag (−1, 0, . . . , 0, 1)


Summation-by-parts Operators

Telescoping flux form

fx(q) = Df + Tp+1 = P−1∆f + Tp+1, Tp+1 = trunc. error

∆ =

−1 1 0 0 0 00 −1 1 0 0 0

0 0. . .

. . . 0 00 0 0 −1 1 00 0 0 0 −1 1

→ [N × (N + 1)]

∆: calculates undivided difference of adjacent fluxes

x1 x2 x3 x4 x5

x0 x1 x2 x3 x4 x5

u0 u1 u2 u3 u4

f1 f2 f3 f4 f5

f0 f1 f2 f3 f4 f5

−1 −910

−√

37

−1645

0−1645

+1645

+√

37

+910

+1


Semi-discrete system

Semi-discrete Navier–Stokes equations

SBP telescoping form

qt = P−1xi

∆xi

(−f(I )

i + f(V )i

)+ P−1

xi(g(B) + g(In))

q(x , 0) = g(0)(x), x ∈ Ω

g(B) contains BC data

g(In) interface penalty

Semi-discrete entropy estimate (1D):

1>PSt = w>B c11Dw − 1>∆F− (Dw)> P c11 (Dw)

Continuous result

d

dt

∫

ΩS dx =

[w>f

(V )i − Fi

]∂Ω−∫

Ωw>xi cij wxj dx



High order entropy stable inviscid flux

f(S)i =

N∑

k=i+1

i∑

`=1

2Q(`,k)fS (q`, qk) , 1 ≤ i ≤ N − 1

Two-points entropy conservative flux of Ismail and Roe for fS (q`, qk)

Entropy stable viscous wall boundary conditions

g(B)1 = −

[f

(I )1 (q)− fsr1

(q, g(E)

)]+[f

(V )1 − f

(V ,B)1

]+M

[w − g(NS),Vel

].

No Penetration Thermal No Slip

Inviscid penalty: flip the sign of the normal velocity component

Viscous flux penalty: prescribe entropy flow = Tx1/T ; BC on T or∇T ; scales with Re

Interior penalty: no-slip condition on the velocity; scales with Re



Entropy Conservative → Stable: A comparison approach

Entropy Conservative Scheme for 3D Euler in hand . . .I What about shocks?I Need dissipation

A Comparison Approach (e.g., the E-Flux condition of Osher)I Two SBP schemes of same orderI Entropy Conservative Scheme + a primary scheme

Both are implemented in telescoping flux formI Compare point-wise flux of primary with entropy stable fluxI Dissipate locally if primary is “less dissipative” than entropy stable flux

f(SSNDG)i = f

(NDG)i + δ

(f

(S)i − f

(NDG)i

), δ =

√b2 + c2 − b√b2 + c2

b = (wi+1 − wi )T(f

(S)i − f

(NDG)i

), c = 10−12


Implementation / Applications

Euler vortex test case: Design Order

Grid L2 error L2 rate L∞ error L∞ rate Cost Ratio

p = 4

009x009 3.37E-08 3.33E-07 1.81017x017 1.68E-09 4.32 2.21E-08 3.91 1.77033x033 5.14E-11 5.03 7.26E-10 4.93 1.56065x065 1.65E-12 4.95 2.18E-11 5.05 1.61129x129 7.48E-13 1.14 4.16E-11 0.92 1.59

Viscous shock test case: Design order with entropy correction

Grid L2 error L2 rate L∞ error L∞ rate Cost Ratio

p = 4

008x008 3.71E-5 8.20E-4 -016x016 1.70E-6 -4.44 5.73E-5 -3.83 1.41032x032 5.70E-8 -4.89 1.79E-6 -5.00 1.43064x064 1.88E-9 -4.92 8.25E-8 -4.43 1.39



Shock Tube Problems: Surprises!

X

Den

sity

0 0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

ExactP3: 128 ElemsESWENO4: 512pts

Entropy correction, t = 0.2

XD

ensi

ty

0 0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

1

ExactP3: 128 Elems NoCorrP9: 32 Elems

No entropy correction, t = 0.2



Sine-Shock Test Problem: Surprises!

X

Den

sity

0 2 4 6 80.5

1

1.5

2

2.5

3

3.5

4

4.5

5 ExactP3: 128 ElemsESWENO4: 512pts

Entropy correction, t = 1.8

XD

ensi

ty

0 2 4 6 80.5

1

1.5

2

2.5

3

3.5

4

4.5

5 ExactP3: 128 Elems NoCorrP9: 32 Elems

No entropy correction, t = 1.8



Accuracy study of the solid wall BCs

3D square cylinder at Re∞ = 200, M∞ = 0.1

X

Y

Z X

Y

Z

t = 1

Grid 1: 3, 024 hexahedrons

Grid 2: every 2nd point Grid 1

Grid 3: every 2nd point Grid 2

u1 p = 1 p = 2 p = 3 p = 4L∞ error rate L∞ error rate L∞ error rate L∞ error rate

Grid 1 4.73e-2 - 2.15e-2 - 9.61e-3 - 4.54e-3 -Grid 2 1.47e-2 1.69 2.88e-3 2.90 5.83e-4 4.04 1.39e-4 5.02Grid 3 3.55e-3 2.04 3.66e-4 2.98 3.40e-5 4.10 4.52e-6 4.94

Solution St 〈cD〉 cRMSL

SSDC p = 1 0.134 1.29 0.12SSDC p = 2 0.154 1.37 0.19SSDC p = 3 0.159 1.40 0.22SSDC p = 4 0.159 1.42 0.23

Sohankar et al. 0.160 1.41 0.22Matteo Parsani 15 / 18


3D Square cylinder: Re∞ = 104, M∞ = 1.5, 4th-order

Grid

1

2

3

Mach

0

3.74

t = 1.5


Conclusions

Conclusions

Interior operator: nonlinearly stable for any diagonal-norm SBPoperators

No-slip wall BC’s: nonlinearly stable and improve dramatically therobustness

Comparison approach allows blending conventional SBP and entropyconservative operators

Noteworthy robustness for shocks even without additional dissipation,limiting, de-aliasing, or ad hoc stabilization techniques

Entropy stable schemes offer possibility of radical advances


Conclusions

Thank you for your attention!

M. H. Carpenter, T. C. Fisher, E. J. Nielsen and S. H. Frankel. Entropy stable spectralcollocation schemes for the Navier–Stokes equations: Discontinuous interfaces. NASATM 218039, September 2013.

M. H. Carpenter, T. C. Fisher, E. J. Nielsen and S. H. Frankel. Entropy stable spectralcollocation schemes for the Navier–Stokes equations: Discontinuous interfaces. SIAM J.Sci. Comput.. In press.

M. Parsani, M. H. Carpenter, E. J. Nielsen. Entropy stable wall boundary conditions forthe compressible Navier–Stokes equations. NASA TM 218282, June 2014.

M. Parsani, M. H. Carpenter, E. J. Nielsen. Entropy stable wall boundary conditions forthe three-dimensional compressible Navier–Stokes equations. Submitted to J. Comput.Phys., June 2014.

M. Parsani, M. H. Carpenter. Entropy stable interior interface coupling for thecompressible Navier–Stokes equations. Submitted to J. Comput. Phys., July 2014.


towards robust, high order and entropy stable algorithms ... · barcelona, spain july 22, 2014....

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