entropy stable high-order schemes for systems of ...entropy stable high-order schemes for systems of...

Entropy stable high-order schemes for systems ofconservation laws.

Siddhartha Mishra

Center of Mathematics for Applications (CMA),University of Oslo, Norway, (and)

Seminar for Applied Mathematics (SAM),ETH Zurich, Switzerland.

Siddhartha Mishra High order entropy stable schemes

Joint work with

I PART 1: Entropy stable FV schemes:I Ulrik S. Fjordholm (SAM, ETH Zurich).I Eitan Tadmor (CSCAMM, U. Maryland, USA.)I Aziz Madrane (Bombardier Aerospace, Montreal, Canada.)

I PART 2: Entropy stable STDG schemes:I Andreas Hiltebrand (SAM, ETH Zurich).

I PART 3: Small scale dependent shock waves.I Phillipe LeFloch (U. Paris VI, France).I M. Castro, C. Pares (U. Malaga, Spain).I L. V. Spinolo (U. Zurich, Switzerland).


Systems of conservation laws

I Systems of conservation laws:

Ut + div(F(U)) = 0.

I Examples:I Shallow water equations (Geophysics)I Euler equations (Aerodynamics)I MHD equations (Plasma physics)


Mathematical Framework

I Shock waves ⇒ Weak (distributional) Solutions

I Uniqueness is an issue: (Entropy conditions).

I Standard paradigm for numerical schemes.


Ingredient I: Approximate Riemann solvers

I Finite volume scheme: ddt Uj(t) + 1

∆x (Fj+1/2 − Fj−1/2) = 0.

X j 1 Xj +1

t n

tn+1

Unj U

nj+1U

nj −1

Un+1j

Fj +1/2Fj −1/2

− 2/ /2

I Numerical flux Fj+1/2 : (approximate) Riemann solver.I Godunov (Roe).I HLL type.


Ingredient II: Non-oscillatory reconstructions

I Piecewise polynomial reconstructions ⇒ Higher order ofaccuracy.

XJ−1/2

XJ+ 1/2

VJ

+

VJ+ 1

−

VJ

JV

+ 1

I Non-Oscillatory reconstructions (control in BV ):I TVD limiters (Van Leer).I ENO reconstruction (Harten et. al.).I WENO reconstruction (Shu, Osher).I DG method (Cockburn,Shu).


Ingredient III: Time stepping

I Strong stability preserving (SSP) Runge-Kutta methods.I Gottlieb, Shu, Tadmor.

I Control in BV .


Standard Paradigm

I Highly succesful in practice.

I Ex: Waves in the sun (Fuchs,McMurry,SM,Waagan):

I Problems still remain !!!


Problem I: Lack of rigorous stability/convergence results

I Stability: continuous framework.I (Multi-dimensional) Scalar conservation laws:

I Existence: BV estimates.I Maximum principles.I Uniqueness: (Infinitely many) Entropy inequalities.


Systems of conservation laws: Entropy framework

I Consider 1-D system: Ut + Fx = 0.

I Assume there exist S (Convex), V and Q with

V∂U = ∂S , V∂F = ∂Q.

I Entropy identity for smooth solutions:

St + Qx = 0.

I Entropy dissipation at shocks ⇒

St + Qx” ≤ ”0.

I provides stability estimate:

d

dt

∫S(U)dx ≤ 0⇒ ‖U(., t)‖L2 ≤ C .

I Holds for several space dimensions.


Robust numerical scheme for conservation laws

• shouldI Entropy stable for non-linear systems.

I Discrete entropy inequalities.

I Convergent forI Linear symmetrizable systems.I Scalar conservation laws.


Existing globally stable schemes

I Scalar equationsI Monotone schemes (1st-order).• Harten, TVD bounds , Crandall, Majda, Entropy estimate.

I TVD limiter based schemes (2nd-order)• Sweby, VanLeer, BV bounds, Osher, Tadmor, Entropybounds.

I ENO schemes (arbitrary order)• No stability bounds, E.Tadmor’s talk

I WENO schemes (arbitrary order)• No (global) stability bounds.

I DG schemes (arbitrary order)• Cockburn, Shu, 1990., BV stability.

I Systems of equations,I Finite volume schemes, Tadmor, 1987, 2003,• Entropy stable Ist order FV scheme


Problem II: Observed numerical instabilites

I Strong shocks:

−1 −0.5 0 0.5 10

5

10

15

ReferenceECERoeRoe

I Vortex Dominated flows:

I Hypersonic flows:


Problem III: Small scale dependent shock waves

I Limit solutions of the hyperbolic-parabolic system:

Uεt + div (F(Uε)) = εdiv (B(Uε)∇Uε)

I Depend explicitly on B: limε→0

Uε,B = UB

I Failure of standard schemes: Boundary value problems

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

1

1.5

2

2.5

3

EDDY

ROE(100 pts)

ROE(1000 pts)


Problem III: Small scale dependent shock waves

I Non-conservative hyperbolic system: Ut + AUx = 0.

I Ex:Multi-layer shallow-water, Coupled Burgers:

−2 0 2 4 6 8 100

2

4

6

8

−2 0 2 4 6 8 100

5

10

15

Godunov

Exact

I Subtle interaction of entropy with viscosity mechanisms.

I Standard paradigm 7→ New paradigm.

I Basis of new paradigm: Entropy stability.


Ingredient I: Entropy conservative fluxes

I Consider the one-d conservation law: Ut + F(U)x = 0.

I Let S be entropy function, flux, define,

V = ∂US , Ψ = 〈V ,F〉 − Q,

I Then

〈V,U〉t + 〈V,Fx〉 = 0,

St + (〈V,F〉)x − 〈Vx ,F〉 = 0,

I Using Ψx = 〈Vx ,F〉, we get that

St + Qx = 0.

I Entropy identity for smooth solutions.


Finite difference scheme

I Conservative semi-discrete scheme:

d

dtUj(t) +

1

∆x(Fj+1/2 − Fj−1/2) = 0

I On the grid:

X j 1 Xj +1

t n

tn+1

Unj U

nj+1U

nj −1

Un+1j

Fj +1/2Fj −1/2

− 2/ /2


Entropy conservative flux: Tadmor,1987

I Assuming〈[[Vj+1/2]],F∗j+1/2〉 = [[Ψj+1/2]].

I We can mimic calculations and arrive at,

d

dtS(Uj(t)) = − 1

∆x(Qj+1/2 − Qj−1/2),

I Existence (Tadmor,1987):

F∗j+1/2 =

∫ 1/2

−1/2F(Vj+1/2(ξ)

)dξ.

I Recent explicit solutions increase computational efficiency:I Shallow water: Fjordholm, Mishra, Tadmor, 2009.I Euler: Roe, 2007.


High-order entropy conservative fluxes

I 2p-th order accurate finite difference generalization (LeFloch,Mercier, Rohde, 2001):

Fp,∗j+1/2 :=

p∑i=1

αpi F∗(Uj ,Uj+i ),

I Fourth-order entropy conservative flux:

F4,∗j+1/2 =

4

3F∗(Uj ,Uj+1)− 1

6(F∗(Uj−1,Uj+1) + F∗(Uj ,Uj+2))

FOURTH ORDER

SECOND ORDER


Comparison

Standard paradigm New paradigm

Approximate Riemann solvers Entropy conservative flux

BV reconstructions

SSP-RK time stepping


Computed heights, 2nd Order scheme

−1 −0.5 0 0.5 1

1.4

1.6

1.8

2

2.2

(a) EEC, 400 pts

−1 −0.5 0 0.5 1

1.4

1.6

1.8

2

2.2

(b) EEC,1600 pts


Oscillations

I Entropy preserving schemes at shocks.

I Oscillations at mesh scale.

I Entropy should be dissipated at shocks.

I Absence of dissipation mechanism ⇒ energy re-distribution tosmallest resolvable scales.

I Have to add Numerical diffusion.


Entropy stable fluxes (Fjordholm, SM, Tadmor, 2009)

I Add dissipation in terms of entropy variables.

I ERoe:

Fj+1/2 = F∗j+1/2 −1

2Rj+1/2|Λj+1/2|R>j+1/2[[Vj+1/2]].

I ERus:

Fj+1/2 = F∗j+1/2−1

2max{|λmax

j |, |λmaxj+1 |}Rj+1/2R

>j+1/2[[Vj+1/2]].

I Resulting semi-discrete schemes are entropy stable.

I Schemes are only first-order accurate.


Comparison


Approximate Riemann solvers Entropy conservative flux+Num diff in entropy var

BV reconstructions



Shallow water Dam break

−1 −0.5 0 0.5 11.4

1.6

1.8

2

ReferenceECRoeRoeRusanov


Normalized run times

Relative error 1 0.5 0.1

Rusanov 1.05 8.24 203.41Roe 1.15 8.43 208.29ERoe 1 7.36 171.7


A different dam-break problem: Strong shock

−1 −0.5 0 0.5 10

5

10

15

ReferenceECERoeRoe


Very-high order numerical diffusion operator

I For formal order ∆xk :

I Replace 〈Dj+1/2,Vj+1 − Vj〉 7→ 〈Dj+1/2,V−j+1 − V+

j 〉I Based on piecewise polynomial reconstruction of order (k − 1)

in each cell.

XJ−1/2

XJ+ 1/2

VJ

+

VJ+ 1

−

VJ

JV

+ 1

I Entropy stability needs to be ensured.


Ingredient III: Entropy stable reconstruction

I Formulated by Fjordholm, SM, Tadmor, 2011.

I Assumes existence of diagonal scaling matrix B ≥ 0 such that

〈V〉j+1/2 := (R>)−1j+1/2Bj+1/2R

>j+1/2[[V]]j+1/2.

⇒ Entropy stability.I Can be ensured if reconstruction

I is in scaled entropy variables: W = R>V.I preserves SIGN PROPERTY (componentwise)

sign(〈w〉j+1/2) = sign([[w ]]j+1/2).


Piecewise linear reconstructions

I Second-order of accuracy.

I Sign property for standard TVD limiter:

Limiter Sign property

Minmod YesSuperbee NoMC NoVan-Leer No

I Higher than second-order accuracy ??


ENO reconstruction: E. Tadmor’s lecture

I Theorem (Fjordholm, SM, Tadmor, 2011):• ENO reconstruction preserves the sign property at any order.

−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2

0

0.2

0.4

0.6

0.8

1

1.2

k = 4

Cell averages

Reconstruction


Arbitrary order entropy stable schemes

I TeCNO scheme:

d

dtUj(t) +

1

∆x(Fj+1/2 − Fj−1/2) = 0

I Numerical flux:

Fj+1/2 = Fp,∗j+1/2 −

1

2Dj+1/2〈V〉j+1/2.

I Entropy conservative flux:

Fp,∗j+1/2 :=

p∑i=1

αpi F∗(Uj ,Uj+i ),

I Explicit two-point entropy conservative flux F∗

I ENO reconstruction in scaled entropy variables.


Arbitrary order entropy stable schemes

I Theorem: Consider system Ut + Fx = 0 with entropy functionS . Then the arbitrary order TeCNO scheme satisfies a discreteentropy inequality:

d

dtSj +

1

∆x

(Qj+1/2 − Qj+1/2

)≤ 0.

and is entropy stable (independent of the order of thescheme).

I Theorem: If the system is linear symmetrizable, then thearbitrary order TeCNO schemes converges weakly in L2 to theunique solution.

I Straightforward to extend on multi-D on Logically rectangulargrids.


Comparison



BV reconstructions Sign property preserving recon



Wave equation: convergence for Sine wave

102

103

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Number of grid points

L1 e

rro

r in

h

Errors for wave equation with u0(x)=sin(4π x). Errors at t=1.

RusENO3

ERusENO3

RusENO4

ERusENO4

RusENO5

ERusENO5


Euler: Shock-turbulence interaction

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

rho

(c) ENO3

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

rho

(d) TeCNO3


Euler: Shock-turbulence interaction

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

rho

(e) ENO4

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

rho

(f) TeCNO4


Advection of Euler vortex: TeCNO2

2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95


Advection of Euler vortex:TeCNO3

2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95


Advection of Euler vortex:TeCNO4

2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95


Advection of Euler vortex

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(g) TeCNO2

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(h) TeCNO3

0 2 4 6 8 10

0.5

0.6

0.7

0.8

0.9

1

(i) TeCNO4


Euler: Cloud-Shock interaction: TeCNO2


Extension to Unstructured meshes: Fjordholm, Madrane,SM, Tadmor.

I Entropy conservative flux: Fij = F1ijn

1ij + F2

ijn2ij

I Components satisfy:

[[V]]>ij Fkij = [[ψk ]]ij k = 1, 2,

I Entropy stable flux:

Fij = Fij −1

2Dij [[V]]ij .

I Example of Numerical diffusion operator:

Dij = R(nij )|Λ(nij )|R>(nij )

.

I First-order accurate.


Unstructured grids

U

Ui

j


Second-order numerical diffusion operator: Fjordholm, SM

I Reconstruction has to satisfy Sign property at edge midpoints.I Modification of the Barth-Jesperson limiter.

I Numerical experiments ongoing.


Outstanding issues

I Very-high order discretizations on unstructured meshes.I Sign property at quadrature points.I Vertex centered ENO formulation (Abgrall).

I Fully discrete high-order schemes:I Numerical experiments indicate SSP-RK3 is entropy stable.


Possible solution to both problems

I Space-time Discontinuous Galerkin methods.I Hiltebrand, SM, in progress.

I Combines ingredients due toI Johnson, Szepessy, Hansbo.I Hughes, Franca, Mallet.I Barth.


Space time element

n

v−v+

vn−

vn+1−


Shock capturing space time DG for ut + (f i(u))xi= 0

I DG formulation: BDG (v,w) + BSD(v,w) + BSC (v,w) := 0.

I DG quasiliner form:

BDG (vh,w) := −∑K ,n

∫K

∫ tn+1

tn

u(vh)wt + f i (vh)wxi dxdt

+∑n,K

∫K

(u(vh(tn+1

− ))w(tn+1− )− u(vh(tn

−))w(tn+))

dx

+∑n,K

∫∂K

∫ tn+1

tn

F∗(vh(x−), vh(x+), ν)w(x−)dσ(x)dt

−∑n,K

∫∂K

∫ tn+1

tn

1

2D(vh(x+)− vh(x−))w(x−)dσ(x)dt.

I DOFs are entropy variables.


Too little diffusion ⇒ Oscillations

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Shock capturing space time DG for ut + (f i(u))xi= 0

I Streamline diffusion quasilinear form:

BSD(vh,w) :=

h∑K ,n

∫K

∫ tn+1

tn

uv(vh)wt + f iv(vh)wxi D(u(vh)t + f i (vh)xi

)dxdt

I Shock capturing operator:

BSC (vh,w) :=

h∑K ,n

∫K

∫ tn+1

tn

‖u(vh)t + f i (vh)xi‖‖∇v‖+ ε

uv(vh)∇vh · ∇wdxdt.


Properties: Hiltebrand, SM.

I Entropy stability of BI Arbitrary order accuracy.

I Maximum principle for scalar conservation laws.

I Convergence to Linear symmetrizable systems.

I Preliminary numerical results.


Convergence rates for smooth solutions: Wave

101

102

103

10−6

10−4

10−2

100

Nx

||u−

ue

xa

ct||

1/||u

exa

ct||

1

deg=0

deg=1

deg=2

deg=3

deg=4


Comparison of different orders:Burgers

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

u

deg=0

deg=1

deg=2

deg=3

exact


Comparison of different schemes:Burgers

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

no SD/SC

SD

SD+SC

exact


Comparison of different schemes: Euler Sod shock tube

−5 0 50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

ρ

no SD/SCSDSD+SC(p)exact


Comparison of different orders: Euler shock tube

−5 0 50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

x

ρ

deg=0deg=1deg=2deg=3exact


Comparison of different orders: Euler Shu-Osher

−5 0 50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x

ρ

deg=0deg=1deg=2deg=3reference


Space time DG: Issues

I Choice of parameters.

I Positivity for systems (a la Zhang, Shu).

I Solutions of the resulting non-linear systems.

I Efficient Preconditioners.


Comparison



BV reconstructions Sign property preserving recon (SG)Shock-capturing STDG (UG)

SSP-RK time stepping SSP-RK time stepping ??(Implicit) space-time DG.


Comparison

High-order (RK) FV Space-time DG

No tuning Some tuningFast (Relatively) SlowStructured grids Unstrutured GridsOne time scale Multiple time scales(Difficult) space time adaptivity space-time adaptivity.Single processor Massively parallel platforms


Small scale dependent shock waves

I Limit solutions of the hyperbolic-parabolic system:

Uεt + div (F(Uε)) = εdiv (B(Uε)∇Uε)

I Depend explicitly on B: limε→0

Uε,B = UB

I Failure of standard schemes: boundary layers

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

1

1.5

2

2.5

3

EDDY

ROE(100 pts)

ROE(1000 pts)


Role of equivalent equation: LeFloch, SM, 2009

I Equivalent equation for scheme:

d

dtUi (t) +

1

h

(Fi+1/2(t)− Fi−1/2(t)

)= 0

I

Ut + Fx = h(B(U)Ux

)x

+R, R =∞∑

q=2p+1

C 2pq (h)q−1

q![Fq(U)]q,

I For standard schemes: B 6= B !!!I New approach: F = F∗ − 1

2D[[V]]i+1/2,I Entropy conservative fluxI Numerical diffusion operator: D = BUV.I Entropy stable scheme with correct equivalent equation (at

leading order).


Boundary value problems: SM, Spinolo, 2011

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Reference

Roe

CND


Non-conservative hyperbolic systems: Castro, Fjordholm,SM, Pares, 2011

−2 0 2 4 6 8 100

5

10u

−2 0 2 4 6 8 100

5

10

15

v

ESPC

Exact

Godunov

ESPC

Exact

Godunov


Issues

I Residual in equivalent equation large at strong shocks

I

Ut + Fx = h(B(U)Ux

)x

+R, R =∞∑

q=2p+1

C 2pq (h)q−1

q![Fq(U)]q,

I High-order schemes reduce the residual LeFloch, SM,forthcoming.

I Modified shock capturing operators in space-time DG.


Comparison


Approximate Riemann solvers Entropy conservative flux+Num diff in entropy var +Matches physical viscosity

BV reconstructions Sign property preserving recon (SG)Shock-capturing STDG (UG)

SSP-RK time stepping SSP-RK time stepping ??(Implicit) space-time DG.


entropy stable high-order schemes for systems of ...entropy stable high-order schemes for systems of...

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