Approximation des systemes hyperboliques par elements ... · Burgers’ equation: f(u)=1 2 u2 Trac ﬂow equation: f(u)=vmax(1 u umax)u Bucley-Leverett: f(u)= u 2 u 2+a(1u), a is

Hyperbolic systems FE approximation Hyperbolic systems + ALE Maximum wave speed

Approximation des systemes hyperboliques par elementsfinis continus non uniformes en dimension quelconque

Jean-Luc Guermond and Bojan Popov

Department of MathematicsTexas A&M University

Seminaire du Laboratoire Jacques-Louis LionsUPMC

04 Nov 2016


Acknowledgments

Collaborators:Murtazo Nazarov (Uppsala University)Vladimir Tomov (LLNL)Young Yang (Penn State University)Laura Saavedra (Universidad Politecnica de Madrid)

Support:


Hyperbolic systems

Hyperbolic systems

1 Hyperbolic systems2 FE approximation3 Hyperbolic systems + ALE4 Maximum wave speed


Hyperbolic systems

The PDEs

Hyperbolic system

@t

u+r·f(u) = 0, (x, t) 2 D⇥R+.

u(x, 0) = u0(x), x 2 D.

D open polyhedral domain in Rd .

f 2 C

1(Rm;Rm⇥d ), the flux.

u0, admissible initial data.

Periodic BCs or u0 has compact support (to simplify BCs)


Hyperbolic systems

Examples

Linear transport: f(u) = �u

Burgers’ equation: f (u) = 12u

2

Tra�c flow equation: f (u) = vmax(1�

u

umax)u

Bucley-Leverett: f (u) = u

2

u

2+a(1�u)2, a is constant ⇠ 1

Shallow water:

u =

✓⇢q

◆2 R1+d , f(u) =

qT

1⇢q⌦q+ 1

2g⇢2I

d

!2 R(1+d)⇥d

p-system

Euler equations

...


Formulation of the problem

Assumptions

9 admissible set A s.t. for all (ul

, ur

) 2 A the 1D Riemann problem

@t

v + @x

(n·f(v)) = 0, v(x , 0) =

(ul

if x < 0

ur

if x > 0.

has a unique “entropy” solution u(ul

, ur

)(x , t) for all n 2 Rd , knk`2 = 1.

There exists an invariant set A ⇢ A, i.e.,

u(ul

, ur

)(x , t) 2 A, 8t � 0, 8x 2 R, 8ul

, ur

2 A.

A is convex.


Formulation of the problem

Examples of invariant sets

Invariant domains are convex for genuinely nonlinear systems (Ho↵ (1979, 1985),Chueh, Conley, Smoller (1973)).

Scalar conservation in Rd : A = [a, b], 8a b 2 R .

Euler: A = {⇢ > 0, e > 0, s � a}, 8a 2 R , where s is the specific entropy.

p-system (1D): etc. U = (v , u)T

A := {U 2 R+⇥R | a W2(U) W1(U) b}, 8a b 2 R

where W1 and W2 are the Riemann invariants

W1(U) = u +

Z 1

v

p�p

0(s) ds, and W2(U) = u �

Z 1

v

p�p

0(s) ds.


FE approximation

Hyperbolic systems



Approximation (time and space)

FE space/Shape functions

{T

h

}

h>0 shape regular conforming mesh sequence

{'1, . . . ,'I

}, positive + partition of unity (P

j2{1: I} 'j

= 1)

Ex: P1, Q1, Bernstein polynomials (any degree)

m

i

:=RD

'i

dx, lumped mass matrix (mi

=P

j2I(Si

)

RD

'i

'j

dx)



Algorithm: Galerkin

Set uh

(x, t) =P

I

j=1 Uj

(t)'j

(x).

Galerkin + lumped mass matrix

m

i

@Ui

@t+

Z

D

r·(f(uh

))'i

dx = 0

Algorithm: Galerkin + First-order viscosity + Explicit Euler

Approximate @Ui

@t byUn+1i

�Un

i

�t

Approximate f(uh

) byP

j2I(Si

)(f(Un

j

))'j

m

i

Un+1i

� Un

i

�t

+

Z

D

r·

0

@X

j2I(Si

)

(f(Un

j

))'j

1

A'i

dx +X

j2I(Si

)

d

n

ij

(Un

i

� Un

j

) = 0.

How should we choose artificial viscosity d

n

ij

?




Introduce

cij

=

Z

D

'i

(x)r'j

(x) dx.

Then

m

i

Un+1i

� Un

i

�t

=X

j

⇣�c

ij

·f(Uj

) + d

n

ij

Uj

⌘.

Observe that conservation impliesP

j

cij

= 0, (partition of unity)

We define d

n

ii

such thatP

j

d

n

ij

= 0, (conservation).

Remark

Rest of the talk applies to any method that can be formalized as above. (FV,

DG, FD, etc.)




Observe that conservation impliesP

j

cij

= 0 andP

j

d

n

ij

= 0.

m

i

Un+1i

� Un

i

�t

=X

j

⇣cij

·(f(Ui

)� f(Uj

)) + d

n

ij

(Ui

+ Uj

)⌘.

Try to construct convex combination . . .

Un+1i

= Un

i

(1 + 2�t

m

i

D

ii

) +X

j 6=i

�t

m

i

⇣cij

·(f(Ui

)� f(Uj

)) + d

n

ij

(Ui

+ Uj

)⌘

= Un

i

(1�

X

j 6=i

2�t

m

i

d

n

ij

) +X

j 6=i

2�t

m

i

d

n

ij

1

2(U

i

+ Uj

) +cij

2dn

ij

·(f(Ui

)� f(Uj

))

!

Introduce intermediate states U(Ui

,Uj

)

U(Ui

,Uj

) :=1

2(U

i

+ Uj

) +cij

2dn

ij

·(f(Ui

)� f(Uj

)).




Now construct convex combination

Un+1i

= Un

i

(1�

X

j 6=i

2�t

m

i

d

n

ij

) +X

j 6=i

2�t

m

i

d

n

ij

U(Ui

,Uj

)

Are the states U(Ui

,Uj

) good objects?




Define nij

= cij

/kcij

k`2 2 Rd , (unit vector).

fij

(U) := nij

·f(U) is an hyperbolic flux by definition of hyperbolicity!

Then

U(Ui

,Uj

) :=1

2(U

i

+ Uj

) +kc

ij

k`2

2dn

ij

(fij

(Ui

)� fij

(Uj

)).



Lemma (GP (2015))

Consider the fake 1D Riemann problem!

@t

v + @x

(nij

·f(v)) = 0, v(x , 0) =

(U

i

if x < 0

Uj

if x > 0.

Let �max(f, nij

,Ui

,Uj

) be maximum wave speed in 1D Riemann problem

Then U(Ui

,Uj

) =R 1

2

� 12

v(x , t) dx with fake time t =kc

ij

k`2

2dnij

, provided

kcij

k`2

2dn

ij

�max(f, nij

,Ui

,Uj

) = t�max(f, nij

,Ui

,Uj

) 1

2

Define viscosity coe�cient

d

n

ij

:= �max(f, nij

,Ui

,Uj

)kcij

k`2 , j 6= i .



Theorem (GP (2015))

Provided CFL condition, (1� 2�t

m

i

|D

ii

|) � 0.

Local invariance: Un+1i

2 Conv{U(Un

i

,Un

j

) | j 2 I(Si

)}.

Global invariance: The scheme preserves all the convex invariant sets.

(Let A be a convex invariant set, assume U0 2 A, then Un+1i

2 A for all n � 0.)

Discrete entropy inequality for all the entropy pairs (⌘, q):

m

i

�t

(⌘(Un+1i

)� ⌘(Un

i

)) +

Z

D

r·(⇧h

q(unh

))'i

dx +X

i 6=j2I(Si

)

d

ij

⌘(Un

j

) 0.



Is it new?

Loose extension of non-staggered Lax-Friedrichs to FE.

Similar results proved by Ho↵ (1979, 1985), Perthame-Shu (1996), Frid (2001)in FV context and compressible Euler.

Some relation with flux vector splitting theory of Bouchut-Frid (2006).

Not aware of similar results for arbitrary hyperbolic systems and continuous FE.


A priori error estimate for scalar equations: Definition of mollifiers

Let � > 0 and ✏ = kfkLip �

Consider mollifiers !� and !✏

!�(t) :=

8><

>:

13� |t| �,2��|t|3�2

� |t| 2�,

0 otherwise,

!✏(x) := ⇧d

l=1!✏(xl

), x := (x1, . . . , xd

).

Following Kruskov (1970), define

�(x, y, t, s) := !✏(x� y)!�(t � s), 8(y, s) 2 D⇥[0,T ].

Following Cockburn-Gremaud (1996,1998), define

��(t) :=

Zt

0!�(s) ds.


A priori error estimate for scalar equations: A useful lemma

Lemma (Guermond, Popov (2014-15))

Assume u0 2 BV (⌦). Let

eu

h

: D⇥[0,T ] �! R be any approximate solution. Assume

that there is ⇤ a bounded functional on Lipschitz functions so that 8k 2 [umin, umax],

8 2 W

1,1c

(D⇥[0,T ];R+):

�

ZT

0

Z

D

�|

eu

h

� k|@t

+ sgn(euh

� k)(f(euh

)� f(k))·r �dx dt

+ k⇡h

�(eu

h

(T )� k)⇡h

(·, Th

)�k`1

h

� k⇡h

�(eu

h

(0)� k)⇡h

(·,�h

)�k`1

h

⇤( ),

where k · k`1h

is the discrete L

1-norm and |T � T

h

| ��t, |0� �h

| ��t, � > 0 is a

uniform constant. Then the following estimate holds

ku(·,T )� u

h

(·,T )kL

1(⌦) c ((✏+ h)|u0|BV (⌦) + ⇤⇤)

where ⇤⇤ := sup0tT

Rt

0

RD

⇤(�)dyds��(t)

.

Generalization of results by Cockburn-Gremaud (1996) and Bouchut-Perthame(1998) based on Kruskov (1970), Kuznecov (1976).


A priori error estimate for scalar equations

English translation

Control on all the Kruskov entropies ) Convergence estimate.

Theorem (Guermond, Popov (2014-15))

Assume u0 2 BV and f Lipschitz. Let u

h

be the first-order viscosity solution. Then

there is c0, uniform, such that the following holds if CFL c0:

(i) ku(T )� u

h

(T )kL

1((0,T );L1) ch

12if a priori BV estimate on u

h

.

(ii) ku(T )� u

h

(T )kL

1((0,T );L1) ch

14otherwise.

BV estimate is trivial in 1D (Harten’s lemma).

BV estimate can be proved in nD on special meshes.

Similar results for FV Chainais-Hillairet (1999), Eymard et al (1998)

First error estimates for explicit continuous FE method (as far as we know).


High-order extension

Higher-order in time

Use SSP method to get higher-order in time.

Strong Stability Preserving methods (SSP), Kraaijevanger (1991) (amazingpaper), Gottlieb-Shu-Tadmor (2001), Spiteri-Ruuth (2002) Ferracina-Spijker(2005), Higueras (2005), etc.:



Higher-order in time

The midpoint rule is not SSP

Heun’s method is SSP

w

(1) = u

n +�tL(tn

, un)

w

(2) = w

(1) +�tL(tn

+�t,w (1))

u

n+1 =1

2u

n +1

2w

(2).

SSPRK(3, 3):

w

(1) = u

n +�tL(tn

, un), z

(1) = w

(1) +�tL(tn

+�t,w (1)),

w

(2) =3

4u

n +1

4z

(1), z

(2) = w

(2) +�tL(tn

+ 12�t,w (2)),

u

n+1 =1

3u

n +2

3z

(2).



Remark on SSP

SSP is not about positivity, it is about convexity.

Let U 7�! S�t

(U) be SSP scheme based on Euler step U 7�! E�t

(U), �t �t0,

Let A be a convex set

then

⇣If Euler step E�t

(U) is invariant domain preserving in A

⌘

then⇣SSP step S�t

(U) is invariant domain preserving in A

⌘

SSP methods preserve convex domains that are invariant for forward Euler timestepping (It’s all about convexity).



Higher-order in space: Entropy viscosity

Use entropy viscosity (or something else)

FCT or other limitation (work in progress)


Strong explosion; ent. vis. sol. 1.5 million P2 nodes

(author: Murtazo Nazarov; 1.5 million P2 nodes)


Mach 10 ramp, ent. vis. sol. 1.2 million P2 nodes

(author: Murtazo Nazarov; 1.2 millions P2 nodes)


Mach 3, Step, ent. vis. sol. 2⇥ 105 P1 nodes and viscous sol. 3.25⇥ 105 P1 nodes


Hyperbolic systems + ALE

Hyperbolic systems



ALE formulation

Instead of tracking the characteristics (there are too many), we want to move themesh.

ALE formulation

Let � : Rd

⇥R+ �! Rd be a uniformly Lipschitz mapping(Rd

3 ⇠ 7�! �(⇠, t) 2 Rd invertible on [0, t⇤])

Let vA

(x, t) = @t

�(��1t

(x), t) Arbitrary Lagrangian Eulerian velocity

We are going to use vA

to move the mesh.

Lemma

The following holds in the distribution sense (in time) over [0, t⇤] for every function

2 C

00 (Rd ;R) (with the notation '(x, t) := (��1

t

(x))):

@t

Z

Rd

u(x, t)'(x, t) dx =

Z

Rd

r·(u⌦ vA

� f(u))'(x, t) dx.


Finite elements

Geometric Finite elements

Let (T 0h

)h>0 be a shape-regular sequence of matching meshes.

Reference Lagrange finite element ( bK , bPgeo, b⌃geo) for geometry

Lagrange nodes {

bai

}

i2{1:ngeosh

} and Lagrange shape functions {

b✓geoi

}

i2{1:ngeosh

}

{ani

}

i2{1: I geo} collection of all the Lagrange nodes in the mesh T

n

h

jgeo : T n

h

⇥{1:ngeosh

} �! {1: I geo} geometric connectivity array

Geometric transformation T

n

K

: bK �! K defined by

T

n

K

(bx) =X

i2{1:ngeosh

}

anjgeo(i,K)b✓geoi

(bx).

) Mesh motion controlled by motion of Lagrange nodes


Finite elements

Approximating Finite elements

Reference finite element ( bK , bP, b⌃)}

Shape functions b✓i

(x) � 0,P

i2{1:nsh

}b✓i

(bx) = 1

Finite element spaces

P(T n

h

) := {v 2 C

0(Dn;R); v|K�Tn

K

2

bP, 8K 2 T

n

h

},

Pd

(T n

h

) := [P(T n

h

)]d ,

Pm

(T n

h

) := [P(T n

h

)]m.

{ n

i

}

i2{1: I} global shape functions in P(T n

h

).


Finite elements

The algorithm

Initialization: m0i

:=RRd

n

i

(x) dx uh0 :=

Pi2{1: I} U

0i

0i

2 Pm

(T 0h

)

ALE velocity field given: wn =P

i2{1: I} Wn

i

n

i

2 Pd

(T n

h

),

Mesh motion:an+1i

= ani

+�twn(ani

).

Mass update: (do not use mn+1i

=RD

n+1i

dx !)

mn+1i

= mn

i

+�t

Z

S

n

i

n

i

(x)r·wn(x) dx.

Update approximation field un+1h

mn+1i

Un+1i

� mn

i

Un

i

�t

�

X

j2I(Sn

i

)

d

n

ij

Un

j

+

Z

Rd

r·

✓ X

j2{1: I}(f(Un

j

)� Un

j

⌦Wn

j

) n

j

(x)

◆ n

i

(x) dx = 0,


Finite elements

Definition of dn

ij

Consider flux gnj

(v) := f(v)�v ⌦Wn

j

, j 2 {1: I}

Consider one-dimensional Riemann problem:

@t

v + @x

(gnj

(v)·nnij

) = 0, (x , t) 2 R⇥R+, v(x , 0) =

(Un

i

if x < 0

Un

j

if x > 0.

Define d

n

ij

by

d

n

ij

= max(�max(gn

j

, nnij

,Un

i

,Un

j

)kcnij

k`2 ,�max(gn

i

, nnji

,Un

j

,Un

i

)kcnji

k`2 ).

Note that

�max(gn

j

, nnij

,Un

i

,Un

j

) = max(|�L

(f, nnij

,Un

i

,Un

j

)�Wn

j

·nnij

|,

|�R

(f, nnij

,Un

i

,Un

j

)�Wn

j

·nnij

|).


Conservation and invariant domain property

Theorem (GPSY (2015))

The total mass

Pi2{1: I} mn

i

Un

i

is conserved.

Provided CFL condition, (1� 2�t

mn

i

|D

ii

|) � 0.

Local invariance: Un+1i

2 Conv{U(Un

i

,Un

j

) | j 2 I(Si

)}.

Global invariance. Let A be a convex invariant set, assume U0 2 A, then

Un+1i

2 A for all n � 0. The scheme preserves all the convex invariant sets.

Discrete entropy inequality for any entropy pair (⌘, q)

1

�t

�mn+1

i

⌘(Un+1i

)� mn

i

⌘(Un

i

)� �

X

j2I(Sn

i

)

d

n

ij

⌘(Un

j

)

�

Z

Rd

r·

✓ X

j2I(Sn

i

)

(q(Un

j

)� ⌘(Un

j

)Wn

j

) n

j

(x)

◆ n

i

(x) dx

Corollary (GPSY (2015))

The scheme preserves constant states (Discrete Global Conservation Law (DGCL))


2D Burgers

@t

u +r·( 12u2�) = 0, u0(x) = 1

S

, with � := (1, 1)T, S := (0, 1)2

Figure: Burgers equation, 128 ⇥ 128 mesh. Left: Q1 FEM with 25 contours; Center left: Final Q1

mesh; Center right: P1 FEM with 25 contours; Right: Final P1 mesh.


Nonconvex flux (KPP problem)

@t

u +r·f(u) = 0, u0(x) = 3.25⇡1kxk`2

<1 + 0.25⇡, with f(u) = (sin u, cos u)T

Figure: KPP problem, 128 ⇥ 128 mesh. Left: Q1 FEM with 25 contours; Center left: Final Q1

mesh; Center right: P1 FEM with 25 contours; Right: Final P1 mesh.


Euler

Compressible Euler, 2D Noh problem, � = 53

Initial data

⇢0(x) = 1.0, u0(x) = �

x

kxk`21x 6=0, p0(x) = 10�15.

Q1 P1

# dofs L

2-norm L

1-norm L

2-norm L

1-norm961 2.60 - 1.44 - 2.89 - 1.71 -

3721 1.81 0.52 8.45E-01 0.77 2.21 0.39 1.09 0.6414641 1.16 0.64 4.21E-01 1.01 1.42 0.64 5.15E-01 1.0858081 7.66E-01 0.60 2.10E-01 0.99 9.39E-01 0.59 2.60E-01 0.99231361 5.21E-01 0.56 1.06E-01 0.98 6.33E-01 0.57 1.28E-01 1.02

Table: Noh problem, convergence test, T = 0.6, CFL = 0.2



Figure: Noh problem at t = 0.6, 96⇥96 mesh. From left to right: density field with Q1

approximation (25 contour lines); mesh with Q1 approximation; density field with P1 approximation(25 contour lines); mesh with P1 approximation.



Figure: Density cuts for the 3D Noh problem at t = 0.6.

Figure: 3D Noh problem at t = 0.6. 64 MPI tasks division.


Maximum wave speed

Hyperbolic systems



How to compute local viscosity?

d

n

ij

:= 2�max(f, nij

,Ui

,Uj

)kcij

k`2 , for j 6= i .

�max(f, nij

,Ui

,Uj

) is max wave speed for Riemann problem


Riemann fan for Euler, p = (� � 1)⇢e

Structure of the Riemann problem (Lax (1957), Bressan (2000), Toro (2009)).

Waves 1 and 3 are genuinely nonlinear (either shock or rarefaction)

Wave 2 is linearly degenerate (contact)

wL

= (⇢L

, uL

, pL

), w⇤L

= (⇢⇤L

, u⇤, p⇤), w⇤R

= (⇢⇤R

, u⇤, p⇤), wR

= (⇢R

, uR

, pR

),


Maximum wave speed bound

Euler system, p = (� � 1)⇢e

Given the states U

L

and U

R

, we have

�1 = u

L

�a

L

✓1 +

(p⇤ � p

L

)+p

L

� + 1

2�

◆ 12

< �3 = u

R

+a

R

✓1 +

(p⇤ � p

R

)+p

R

� + 1

2�

◆ 12

where p

⇤ is the pressure of the intermediate state.

Then and define�max(U

L

,UR

) = max(|�1|, |�3|).

In practice we just need a good upper bound of p⇤: p

⇤� p

⇤. Then

�max(UL

,UR

) = max(|�1(p⇤)|, |�3(p

⇤)|).



To avoid computing p

⇤, it is a common practice to estimate �max bymax(|u

L

|+ a

L

, |uR

|+ a

R

)

This estimate is inaccurate and can be wrong.



Counter-example 1: 1-wave and the 3-wave are both shocks Toro 2009, §4.3.3

⇢L

⇢R

u

L

u

R

p

L

p

R

5.99924 5.99242 19.5975 -6.19633 460.894 46.0950

�max ⇡ 12.25 but max(|uL

|+ a

L

, |uR

|+ a

R

) ⇡ 29.97, large overestimation



Counter-example 2: 1-wave is a shock and the 3-wave is an expansion

⇢L

⇢R

u

L

u

R

p

L

p

R

0.01 1000 0 0 0.01 1000

�max ⇡ 5.227 but max(|uL

|+ a

L

, |uR

|+ a

R

) ⇡ 1.183, large underestimation


Definition of p⇤

Let p⇤ be the zero of �R

, then

p

⇤ =

0

B@a

L

+ a

R

�

��12 (u

R

� u

L

)

a

L

p

� ��12�

L

+ a

R

p

� ��12�

R

1

CA

2��1

Lemma (GP (2016))

We have p

⇤ < p

⇤in the physical range of �, 1 < �

53 .

p

⇤ is an upper bound on p

⇤.

min(pL

, pR

) p

⇤ p

⇤ (starting guess for cubic Newton alg., GP (2016))


Conclusions

Continuous finite elements

Continuous FE are viable tools to solve hyperbolic systems.

Continuous FE are viable alternatives to DG and FV.

Continuous FE are easy to implement and parallelize.

Exa-scale computing will need simple, robust, methods.

Current and future work

Convergence analysis, error estimates beyond first-order.

Extension to DG.

Extension of BBZ to higher-order polynomials (order 3 and higher).

Extension of BBZ to systems (Shallow water, Euler).

Extension to equations with source terms (Radiative transport, Radiativehydrodynamics).

Approximation des systemes hyperboliques par elements ... · Burgers’ equation: f(u)=1 2 u2 Trac ﬂow equation: f(u)=vmax(1 u umax)u Bucley-Leverett: f(u)= u 2 u 2+a(1u), a is

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