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
MotivationMethodology
Burgers EquationNavier-Stokes
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
Problems of Interest Large Eddy Simulations
LES requires resolution of large scale flow features Mathematical formalism/uncertainty analysis needed to evaluate
resolution
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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!)
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Problems of InterestLinear Analysis
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Entropy ConsistencyEntropy StabilityWENO
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Entropy ConsistencyEntropy StabilityWENO
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Entropy ConsistencyEntropy StabilityWENO
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)
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Entropy ConsistencyEntropy StabilityWENO
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
)
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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.
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Multi-Element Airfoil; Mach0 = 0.25, Re = 30, 000
Mach ; Vorticity
Dilatation ; Entropy
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Compression CornerAdaptation of Wu and Martin: M = 2.9, Reθ = 2300
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Supersonic Cylinder
Cylinder in a duct with inviscid walls M = 3.5, Re = 10000
Density
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Supersonic Cylinder
Cylinder in a duct with inviscid walls M = 3.5, Re = 10000
Mach Number
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
FUN3D Medium GridFUN3D Fine GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Comparison to FUN3DDensity, x = 5
X
Den
sity
-2 0 2
0.4
0.6
0.8
1
1.2
1.4
1.6
FUN3D Medium GridFUN3D Fine GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
FUN3D Medium GridFUN3D Fine GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
VULCAN Medium GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
Comparison to VULCANDensity, x = 5
X
Den
sity
-2 0 2
0.4
0.6
0.8
1
1.2
1.4
1.6
VULCAN Medium GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
VULCAN Medium GridHOFD Coarse GridHOFD Medium GridHOFD Fine Grid
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes
MotivationMethodology
Burgers EquationNavier-Stokes
Inviscid PartViscous PartStatus
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
NIA 04/24/2012 Entropy Stable HOFD Schemes