entropy stable high order finite difference schemes...

MotivationMethodology

Burgers EquationNavier-Stokes

Entropy Stable High Order Finite DifferenceSchemes for Finite Domain Compressible

Flows

Travis C. Fisher,1 Mark H. Carpenter,1 and Nail K. Yamaleev2

1Computational AeroSciences BranchNASA Langley Research Center, Hampton, VA

2Department of MathematicsNorth Carolina A&T State University, Greensboro, NC

National Institute of Aerospace CFD Seminar04/24/2012

NIA 04/24/2012 Entropy Stable HOFD Schemes



Problems of InterestLinear Analysis

Problems of Interest All Compressible Mach Numbers

Shocks and turbulence represent competing numerical challenges High order algorithms not mainstream Must be able to handle non-trivial geometries





Problems of Interest Laminar and Direct Numerical Simulations (DNS)

Need low numerical noise for transition simulations Need to control artificial dissipation Adaptivity a key need





Problems of Interest Large Eddy Simulations

LES requires resolution of large scale flow features Mathematical formalism/uncertainty analysis needed to evaluate

resolution





Nonlinear Conservation Laws Fundamental forms

Inviscid

ddt

Z xR

xL

q dx + f (q)|xRxL

= 0, x ∈ [xL, xR ], t ∈ [0,∞)

qt + f (q)x = 0, x ∈ [xL, xR], t ∈ [0,∞)

Viscous

qt + f (q)x = f (v)(q)x , x ∈ [xL, xR ], t ∈ [0,∞)

Numerical theory for systems is far from complete Consider problems that admit shocks in inviscid limit Use adaptive (WENO), upwind, or hybrid numerical methods Numerical solution with vanishing viscosity must converge to

appropriate weak solution (need proof!)





Linear Energy Analysis

Assume smooth solutions and change to primitive form

vt + vq fqqv vx = vq f (v)x , x ∈ [xL, xR ], t ∈ [0,∞)

Linearize about constant state v = v + v

vt + Avx = Cvxx

Symmetrize the system with w = R−1v

wt + Awx = Cwxx ,

A = R−1AR = AT C = R−1CR = CT ,

ζT Cζ ≥ 0





Linear Energy Analysis

Contract symmetric variables and integrate

ddt

∫ xR

xL

wT w2

dx = −12

[

wT Aw − 2wT Cwx

]xR

xL

−

∫ xR

xL

wTx Cwx dx

Energy growth rate depends only on boundary data Discrete analog achieved using summation-by-parts operators Stability proof using mimetic argument

Variable coefficient terms Smooth nonlinear problems

Linear mimetic argument is invalid for discontinuous problems





Linear Shortfalls Unexpected behavior at shocks

x

u(x)

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

InitialWENOESWENO





Linear Shortfalls The problem

ut + P−1∆f(W )

= 0, ∆f(W )

=12

(Q + R)Uu,

ut + P−1∆f(ESW )

= 0, ∆f(ESW )

=12

(Q + R + Rc)Uu

Semi-discrete nonlinear energy difference

uT ∆(f(ESW )

− f(W )

) = uTRcUu

ESWENO ensures R + Rc and Rc are semidefinite RcU is indefinite, may lead to energy growth Need schemes that satisfy nonlinear energy directly




Entropy ConsistencyEntropy StabilityWENO

Entropy Analysis Mathematical entropy-entropy flux pair (S, F )

Nonlinear functions of solution, S = S(q), F = F (q) The entropy is convex

ζT Sqqζ > 0 ∀ζ 6= 0

The Hessian of the entropy symmetrizes all coefficient matrices The entropy flux satisfies

Sq fq = Fq

A set of entropy variables is defined, wT = Sq The entropy equation (smooth solutions)

Sqqt + Sq fx − Sq f (v)x = St + Fx − Sq f (v)

x = 0

The entropy inequality (discontinuous solutions)

St + Fx ≤ 0





Entropy Consistent Numerical Methods

Want to exactly mimic integral property of continuous entropy

ddt

∫ xR

xL

Sdx = − F |xRxL

For semidiscrete equation

ut + P−1∆f = 0

The semidiscrete entropy is

wTPut + wT ∆f = 0

Entropy consistent spatial discretizations yield

ddt

wTPu + FN − F1 = 0, wT ∆f = FN − F1





Entropy Stability

In generalddt

∫ xR

xL

Sdx ≤ − F |xRxL

Inequality applies in viscous/discontinuous cases Numerical methods must dissipate energy at shocks Dissipation must be added without violating Lax Wendroff theorem Following Tadmor, entropy consistent is upper bound

Semidiscrete analog:

ddt

wTPu + FN − F1 ≤ 0, wT ∆f ≤ FN − F1

Viscous:

ddt

wTPu + FN − F1 =(

wT f (v))

N−

(

wT f (v))

1− Ξ, Ξ ≥ 0∀w





WENO

ui−1 ui ui+1 ui+2

SL SC SR

fi

Start by discretizing equation in divergence form

ut + P−1Qf = ut + P−1∆f = 0

Recast fluxes, f into a combination of candidate interpolations

f =X

r

d (r)f f (r)

Replace target weights with nonlinear weights

f =X

r

ω(r)f f (r)





ui−1 ui ui+1 ui+2

SL SC SR

fi

Smoothness indicators are based on solution

τ = (ui−1 − 3ui + 3ui+1 − ui+2)2,

βL = (ui − ui−1)2, βC = (ui+1 − ui)

2, βR = (ui+2 − ui+1)

2

Calculate weights

ω(r)f =

γ(r)f

X

r

γf

, γ(r)f = γf

“

d (r)f , τ , β

(r)”

Large values of β result in negligible weight Full formal boundary closures are used




Inviscid PartViscous PartStatus

Burgers Equation

Use Burgers equation as a model problem

qt +

(

q2

2

)

x= µqxx , x ∈ [xL, xR ], t ∈ [0,∞)

Entropy is the same as the nonlinear energy

S =q2

2, F =

q3

3, w = Sq = q, Sqq = 1

Inviscid terms are split Viscous terms are nearly trivial Formal L2 stability proof results





Equation Splitting

For fluxes of the form f (q) = w(q)v(q)

Split using linear combination of divergence and product ruleterms

qt + αfx + (1 − α) (vwx + wvx ) = 0

Discretize using nondissipative summation-by-parts finitedifferences

ut + αDf + (1 − α) (VDw + WDv) = 0

Recast into flux difference form

ut + P−1∆f = 0

Elements of f are consistent with RH flux Lax Wendroff is satisfied for any α





Entropy Stable Inviscid Terms

For Burgers v(q) = q, w(q) = q/2 For α = 2/3 the inviscid terms are entropy consistent

13

uTPP−1 (QU + UQ) u =13

uBUu =13

(

u3N − u3

1

)

We have the additional property

U∆f = ∆F

Shown to satisfy

(uj+1 − uj) f (s)j = ψj+1 − ψj , j = 1, . . . , N − 1,

ψj =u3

j

6, ∀j





Entropy Stable Inviscid Terms

x1 x2 x3 xi xi+1 xNxN−1xN−2

x0x1 x2 x3 xi

xNxN−1xN−2xN−3

fifi−1

xi−1 xi+2

xi−1

This local property is used to define entropy stable interpolatedfluxes

(uj+1 − uj) fj ≤ ψj+1 − ψj

For a dissipative WENO operator, we must ensure

(uj+1 − uj) f (w)j ≤ (uj+1 − uj) f (s)

j

Thus the WENO operator can never add negative dissipation





Overcoming the Shortfalls

Resolves issue at shock

x

u(x)

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

InitialWENOESWENOSSWENO





Overcoming the Shortfalls

As good for boundary closure as ESWENO

x

u(x)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

WENOESWENOSSWENO





Viscous Terms Narrow stencil SBP viscous terms are utilized

ut + P−1∆f = µP−1(R + BS)u

Entropy stability is immediately satisfied

ddt

12

uTPu = −13

(

u3N − u3

1

)

+ µ ((Su)N − S(u)1) + µuTRu,

ζTRζ ≤ 0, ∀ζ

Semidiscrete entropy dissipation rate Design order approximation of continuous Not bounded by continuous case

Much more difficult for variable viscosity





Status Fully implemented with formal boundary closures

Boundary conditions through SAT penalties Entropy stable penalties

L2 stability proved for both continuous and discontinuous. Stability proof maintained with extension to Burgers system No degradation of accuracy for curvilinear grids





Navier-Stokes Start with calorically perfect equations

qt + (fi)xi=

(

f (v)i

)

xi

, x ∈ Ω, t ∈ [0,∞),

f (v)i = cijqxj .

Only one known entropy for Navier-Stokes

S = −ρs, Fi = −ρuis, Sq = wT , ζT Sqqζ > 0, T , ρ > 0

Symmetrizes inviscid and viscous coefficient matrices

qw wt + (fi)w wxi =(

cijwxj

)

xi, (fi)w = (fi)

Tw , cij = cT

ji





Navier-Stokes Contracting with entropy variables

Sqqt + Sq (fi)xi= Sq

(

cijwxj

)

xi

Resulting in the entropy equation

St + (Fi)xi=

(

wT cijwxj

)

xi− wT

xicijwxj

Integrating over space we get the entropy integral

ddt

∫

Ω

S dV = −

∫

∂Ω

(

Fi − wT f (v)i

)

dSi −

∫

Ω

wTxi

cijwxj dV

No simple equation splitting found that can recover entropyconsistency





Semidiscrete Entropy Analysis

Semidiscretization

ut +(

P−1∆)

i f i =(

P−1∆)

i f(v)

i

Discrete entropy decay

wTPΩut = wTP∂Ω∆i

(

f(v)

i − f i

)

Rearranging

ddt

1TPΩS = 1TPΩWut = 1TP∂Ω

(

W∆i f(v)

i − W∆i f i

)





Entropy consistency

Inviscid entropy consistency satisfied if

wT ∆αfα = (Fα)1 − (Fα)Nα

However, as in Burgers we seek

W∆αfα = ∆αFα

Satisfied when

(wj+1 − wj)T

(

f (s)α

)

j= ψj+1 − ψj , j = 1, . . . , Nα − 1

ψj = wTj uj − Fj , ∀j





Entropy consistency Roe’s second order solution (first affordable solution)

A special set of logarithmic averages Non-dissipative Not immediately extensible to thermally perfect/reacting

Lefloch extension to higher resolution Richardson extrapolation type combination of stencils Still only second order in finite volume form Extended herein to finite difference form, recovering high order Extended herein to formal boundary closures

Dissipation still must be added





Entropy Consistency in action

Inviscid Taylor Green Vortex, M = 0.1 Kinetic energy nearly conserved Entropy conserved until discontinuous

Time

TK

E

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

5th Order WENO JS4th Order S-consistent4th Order SSWENO6th Order SSWENO





Entropy Consistency in action

No lower bound on scale A stable solution is not necessarily an accurate solution Most stable form, but still won’t run to T → ∞

Time

TK

E

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

5th Order WENO JS4th Order S-consistent4th Order SSWENO6th Order SSWENO





Entropy Stable WENO

In general we require

(wj+1 − wj)T

(

f (w)α

)

j≤ ψj − ψj , j = 1, . . . , Nα − 1

Thus we have a local limit for the WENO flux

(wj+1 − wj)T

(

f (w)α − f (s)

α

)

j≤ 0

Again ensures WENO does not artificially add energy Improves stability, but not a proof Does not bound density or temperature away from zero In practice, more robust and simpler to implement than ESWENO





Entropy Stable Viscous Terms

Nontrivial to get narrow stencil and entropy stability Requires variable coefficient second derivative operator Second derivative operator must be compatible with first

derivative

D2 = P−1(R(µ) + B[µ]S), R = −DTP[µ]D + M(µ),

ζTM(µ)ζ ≤ 0, µ > 0, ∀ζ

Must use entropy variables in viscous terms

f (v) = cijwxj





Multi-Element Airfoil; Mach0 = 0.25, Re = 30, 000

Grid and Vorticity Contours

Figure: Multi-Block grid with 65 domains, and C0 block:block connectivity.





Multi-Element Airfoil; Mach0 = 0.25, Re = 30, 000

Mach ; Vorticity

Dilatation ; Entropy





Compression CornerAdaptation of Wu and Martin: M = 2.9, Reθ = 2300





Supersonic Cylinder

Cylinder in a duct with inviscid walls M = 3.5, Re = 10000

Density





Supersonic Cylinder

Cylinder in a duct with inviscid walls M = 3.5, Re = 10000

Mach Number





Comparison to FUN3DDensity, y = 0

X

Den

sity

2 4 6 8 10 12

0.2

0.4

0.6

0.8

1

1.2

1.4

FUN3D Medium GridFUN3D Fine GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid





Comparison to FUN3DX Velocity, y = 0

X

X V

eloc

ity

2 4 6 8 10 12-0.4

-0.2

0

0.2

0.4

0.6

0.8






Comparison to FUN3DDensity, x = 5

X

Den

sity

-2 0 2

0.4

0.6

0.8

1

1.2

1.4

1.6






Comparison to FUN3DX Velocity, x = 5

X

X V

eloc

ity

-2 0 2-0.2

0

0.2

0.4

0.6

0.8

1






Comparison to VULCANDensity, y = 0

X

Den

sity

2 4 6 8 10 12

0.2

0.4

0.6

0.8

1

1.2

1.4

VULCAN Medium GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid





Comparison to VULCANX Velocity, y = 0

X

X V

eloc

ity

2 4 6 8 10 12-0.4

-0.2

0

0.2

0.4

0.6

0.8






Comparison to VULCANDensity, x = 5

X

Den

sity

-2 0 2

0.4

0.6

0.8

1

1.2

1.4

1.6






Comparison to VULCANX Velocity, x = 5

X

X V

eloc

ity

-2 0 2-0.2

0

0.2

0.4

0.6

0.8

1






What is it that we’ve done? Entropy stability and consistency yield more robust high order

methods Not a full nonlinear stability proof Still must satisfy linear stability for BC’s First known construction of an entropy stable WENO FD scheme First known application of high order entropy stable viscous terms

Formal boundary closures follow from Burgers analysis Domain interfaces more robust in the presence of shocks





What is left to do? A true stability proof Entropy stable or nonlinear stability for penalties More work on interfaces Extension to multicomponent reacting flows More efficient high order entropy consistent forms


entropy stable high order finite difference schemes...

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