the structure of wb schemes for friedrichs systems with relaxation · 2017. 2. 21. · two-state...

B. Despres(LJLL-

UPMC)with C. Buet

(CEA)

collaborationwith E.

Franck (ex.PhD) and T.Leroy (PhD)

The structure of WB schemes for Friedrichssystems with relaxation

B. Despres (LJLL-UPMC)with C. Buet (CEA)

collaboration with E. Franck (ex. PhD) and T. Leroy (PhD)

Support of CEA is acknowledged p. 1 / 35

Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Stuttgart 2014 p. 2 / 35

Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Comparison standard FV versus WB-FV

Initial stage of Rayleigh-Taylor instability :source term = gravity.

standard FV WB-FVspurious velocities vortex is ”correct”


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Introduction

Consider a given system

∂tU +∇ · f (U) = S(U, . . . )

with standard splitted FV discretization

Un+1j − Un

j

∆t+

f nj+ 1

2

− f nj− 1

2

∆x= S(Un

j , . . . ) (1)

WB-FV scheme = exact for stationary solutionsAP (asymptotic preserving) schemes = WB with stiffness.

Observation : the standard scheme for (1) is not WB.

Literature : strategy is ”one case after another”.

Can we have a global view on WB methods ?

=⇒ Need of a model problem.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Linearization of non linear system with relaxation

A paradigm is the p-system with friction : linearization yields∂tτ − ∂xu = 0,∂tu + ∂xp(τ) = −σu. =⇒

∂tτ1 − ∂xu1 = 0,∂tu1 − c2

0∂xτ1 = −σu1.

Set U = (c0τ1, u1)t . One gets

∂tU + A∂xU = −RU,

where A = −c0Id = At and

R = σ

(0 00 1

)= Rt ≥ 0.

This is the linear hyperbolic heat equation.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Approximation of kinetic equations with Friedrichssystem of large size n >> 1

An example is DOM (Discrete Ordinates Method) for transfer

∂t I + µ∂x I = σ (< I > −I )

Decide weights wi > 0 and directions µi : approximate

I (x , t, µ) =n∑

i=1

wi fi (x , t)δ(µ− µi ) +n∑

i=1

wigi (x , t)δ(µ+ µi ).

After normalization U = (√

diag(wi ) f ,√

diag(wi ) g)t ∈ R2n,one gets the Sn model

∂tU + A∂xU = −RU, R = −w ⊗w + Id ,

where w = (√w1, . . . ,

√wn,√w1, . . . ,

√wn) ∈ R2n with

‖w‖ = 1.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Model problem = Friedrichs systems withrelaxation

• We will consider

∂tU + ∂x(AU) + ∂y (BU) = −RU, U(t, x) ∈ Rn,

where x = (x , y) ∈ R2,and typically A = At , B = Bt and R = Rt ≥ 0.

• The vectorial space of stationary states is

U = x 7→ U(x); ∂x(AU) + ∂y (BU) = −RU .


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Main idea : the dual equation

Def : ∂tV + At∂xV + Bt∂yV = RtV , V (t, x) ∈ Rn.

Property : ∂t(U,V ) + ∂x(AU,V ) + ∂y (BU,V ) = 0.

• The vectorial space of dual stationary states is

V =

x 7→ V (x); At∂xV + Bt∂yV = RtU.

• Pick up Vp ∈ V and define αp = (U,Vp) ∈ R

∂tαp + ∂x(AU,Vp) + ∂y (BU,Vp) = 0


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Strategy

a) Assemble the system of conservation laws for α = (αp).b) Discretize the new (system of) conservation laws.c) Rewrite the new scheme for the original primal variable.

The dimension of V which is the number of independent αp

dim(V) = dim(U)

is extremely important in the discussion.

For simplicity, matrices will be now constant in space.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Section 2

dim(V) = n


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Non singular 1D case : det(A) 6= 0

∂xV = A−tRtV =⇒ V (x) = eA−tRtxV (0) where

eA−tRtx is a matrix exponential : dim(V) = n.

∂t

(U, eA

−tRtxV (0))

+ ∂x

(AU, eA

−tRtxV (0))

= 0, ∀V (0).

∂t (P(x)U,V (0)) + ∂x (P(x)AU,V (0)) = 0, ∀V (0)

Proposition : Define the change-of-basis matrixP(x) = eRA

−1x , and the change of unknown is

α = P(x)U ⇐⇒ U = P−1(x)α

The new conservative system rewrites

∂tα + ∂x(Q(x)α) = 0, Q(x) = P(x)AP−1(x).


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Discretization

• For further simplification, A is also symmetric :A = At ∈ Rn×n.

• The FV discretization of ∂tα + ∂xβ = 0 with β = Q(x)αwrites

αn+1j − αn

j

∆t+βnj+ 1

2

− βnj− 1

2

∆xj= 0

where βnj+ 1

2

is the flux at time step tn = n∆t.

• Notice that∂tβ + Q(x)∂xβ = 0.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Notations for the standard Riemann solver

This is based on the spectral decomposition

Aup = λpup, λp 6= 0.

It writes (up,U

∗j+ 1

2

− Uj

)= 0, λp > 0,(

up,U∗j+ 1

2

− Uj+1

)= 0, λp < 0.

Intermediate state

RL

The intermediate state is computed in function of a left stateL = j and a right state R = j + 1.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Diagonalisation of Q

The right and left spectral decompositionsQ∗ = Q(x∗) = P(x∗)AP(x∗)−1 are

Q(x∗)r∗p = λpr∗p , r∗p = P(x∗)up,

Qt(x∗)s∗p = λps∗p , s∗p = P(x∗)−tup.

Solve locally the system ∂t(s∗p , β

)+ λp∂x

(s∗p , β

)= 0.

One-state solvers : given by the well-posed linear system (s∗p , β

∗ − βL)

= 0, λp > 0,(s∗p , β

∗ − βR)

= 0, λp < 0.

It defines a first Riemann solver ϕ : Rn × Rn × R→ Rn

ϕ(βL, βR , x∗) = β∗.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

The scheme with the one-state solver

The scheme writes

αn+1j − αn

j

∆t+ϕ(βnj , β

nj+1, xj+ 1

2)− ϕ(βnj−1, β

nj , xj− 1

2)

∆xj= 0

or

Un+1j − Un

j

∆t+P(xj)

−1ϕ(βnj , β

nj+1, xj+ 1

2)− ϕ(βnj−1, β

nj , xj− 1

2)

∆xj= 0

Property : the scheme is WB.

Proof : If βj = β for all j , the solution is stationary.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Local formulation

Un+1j − Un

j

∆t+ A

U∗j+ 1

2

− U∗j− 1

2

∆xj

+P(xj)

−1P(xj+ 12)− I

∆xjAU∗

j+ 12

+I − P(xj)

−1P(xj− 12)

∆xjAU∗

j− 12

= 0.

where the ”flux” U∗j+ 1

2

is solution of the linear system

(up,U

∗j+ 1

2

− e−A−1R∆x+

j Uj

)= 0, λp > 0,(

up,U∗j+ 1

2

− e−A−1R∆x−j+1Uj+1

)= 0, λp < 0,

with ∆x±j = xj± 12− xj .

Notice the compatible discretization of the sink.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Schematic

R

Intermediate state

RL

NL

N

Structure of the one-state solver. The New-left and New-rightstates are modification of the initial Left and Right states.

Here all positive (or negative) waves are collapsed in one.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

The two-states solver

Same principle but with upwinding of the eigenvectors (sLp , β

∗∗ − βL)

= 0, λp > 0,(sRp , β

∗∗ − βR)

= 0, λp < 0,

Theorem : Assume R ≥ 0. Then the familysLpλp>0

∪sRpλp<0

is linearly independent.

It defines a second Riemann solver ψ : Rn × Rn × R× R→ Rn

ψ(βL, βR , xL, xR) = β∗∗.

- Imagine the one-state solver as a Roe mean solver, and the two-statessolver as more adapted to multi-material simulations.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Local formulation of the two-states solver

One gets

Un+1j − Un

j

∆t+ A

U∗∗j+ 1

2

− U∗∗j− 1

2

∆xj

+P(xj)

−1P(xj+ 12)− I

∆xjAU∗∗

j+ 12

+I − P(xj)

−1P(xj− 12)

∆xjAU∗∗

j− 12

= 0.

where the ”flux” U∗∗j+ 1

2

is solution of the linear system

(up, e

A−1R∆x+j U∗∗

j+ 12

− Uj

)= 0, λp > 0,(

up, eA−1R∆x−j+1U∗∗

j+ 12

− Uj+1

)= 0, λp < 0,


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Interpretation

R

Intermediate state

RL

NL

N

Schematic of the two-states solver. The New-left andNew-right internal states are modification of the intermediatestate by the backward propagator.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Interpretation for : ∂tp + ∂xu = 0,∂tu + ∂xp = −σu

A−1R =

(0 σ0 0

)is nilpotent. So eA

−1Rx = I + A−1Rx .

One-state solver = Steady state approximation of Huang-Liu1986, or Jin-Levermore 1996 p∗

j+ 12

=pj + pj+1

2+

(1 − σ∆x+j )uj − (1 + σ∆x−j+1)uj+1

2,

u∗j+ 1

2=

pj − pj+1

2+

(1 − σ∆x+j )uj + (1 + σ∆x−j+1)uj+1

2.

Two-state solver= Equal to the Gosse-Toscani scheme 2002.p∗∗j+ 1

2=

1−σ∆x−j+1

2+σ∆x+j −σ∆x−j+1

(pj − uj) +1+σ∆x+

j

2+σ∆x+j −σ∆x−j+1

(pj+1 − uj+1),

u∗∗j+ 1

2= 1

2+σ∆x+j −σ∆x−j+1

(uj + uj+1 + pj − pj+1).

- Immediate interest for non linear fluid solvers.

- To be compared with recent Gosse textbook (Springer 2013) on WB-AP,

but with a completely different localization method.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Section 3

dim(V) < n


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Degenerate case det(A) = 0

• The matrix exponentials P(x) = eRA−1x blow up

=⇒ the previous theory does not make sense.

• Let us consider an example P1 radiation model coupled alinear temperature equation. It writes

∂tp +∂xu = τ(T − p),∂tu +∂xp = −σu,∂tT = τ(p − T ),

(2)

for which

A =

0 1 01 0 00 0 0

and R =

τ 0 −τ0 σ 0−τ 0 τ

.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Assuming that σ, τ > 0, the solutions of the adjoint stationaryequation satisfy

∂x u = 0, ∂x p = σu, T = p.

So dim(V) = 2 < 3 which indicates a degeneracy.Basis functions are

V1 = (1, 0, 1)t and V2 = (σx , 1, σx)t

so one can write only two conservation laws for

α1 = (U,V1) = p + T and α2 = (U,V2) = σx(p + T ) + u.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Ad-hoc time integration

Defining arbitrarily α3 = T , one obtains∂tα1 + ∂x(−σxα1 + α2) = 0,∂tα2 + ∂x

((1− σ2x2)α1 + σxα2 − α3

)= 0,

∂tα3 = τ(α1 − 2α3).

The last equation is more an ODE (e2τ tα3)′ = τe2τ tα1.Elimination over [0,∆t] yields the interesting 2× 2 system

∂tα1 +∂x(−σxα1 + α2) = 0,∂tα2 +∂x

((1− σ2x2)α1 + σxα2

−(1− 1

2e−2τ∆t

)α1 + e−2τ∆tα3(0)

)= 0.

- Numerics and numerical analysis still to be done.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Section 4

dim(V) =∞


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

MultiD : hyperbolic heat equation

• Assume σ > 0 is constant∂tp +∂xu +∂yv = 0,∂tu +∂xp = −σu,∂tv +∂yp = −σv ,

Here det(A) = 0, det(B) = 0 and AB 6= BA.

• The adjoint stationary states (p, u, v) are∂x u +∂y v = 0,∂x p = σu,

∂y p = σv .

So (u, v) = 1σ∇p and ∆p = 0. Therefore

V =

(p, u, v); p is harmonic and (u, v) =

1

σ∇p.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

• We define

Vn = V ∩ p is an harmonic polynomial of degree ≤ n .

• So dim(Vn)→∞ = dim(V)

• p ∈ V1 is equivalent to p = a + bx + cy .It yields three test functions

V1 =

100

, V2 =

σx10

, V3 =

σy01


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Set α1 = p, α2 = σxp + u and α3 = σyp + v .The system writes

∂t

α1

α2

α3

+ ∂x

m1

α1 + σxm1

σym1

+ ∂y

m2

σxm2

1 + σym2

= 0,

(3)where m1 = −σxα1 + α2 and m2 = −σyα1 + α3.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Algebraic miracle

The original variable can be recovered U = P(x)−1α where

P(x) =

1 0 0σx 1 0σy 0 1

is non singular. The inverse formula is P(x)−1 = P(−x). Thecommutation-composition property holds

P(x)P(y) = P(y)P(x) = P(x + y).

The system writes

∂tα + ∂x(P(x)AP(x)−1α

)+ ∂y

(P(x)BP(x)−1α

)= 0.

Many more conservation laws (2 more for V2).


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Some propaganda for nodal solvers : xr is for nodes

Denote the corner normal njr = (cos θjr , sin θjr ).

x j

xr+1

xr−1

xr

Cell Ω j

Cell Ωk

l jrn jr

sjp′j(t) +

∑r ljrnjr · ur = 0,

sju′j(t) +

∑r ljrnjrpjr = 0.

Equivalent to

sjU′j (t) +

∑r

ljrAjrUjr = 0, (4)

where Uj = (pj ,uj), Ujr = (pjr ,ur ) andAjr = cos θjrA + sin θjrB = At

jr is symmetric matrix.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

WB scheme

• Immediate extension of 1D local formulation yields

sjU′j (t) +

∑r

ljrAjrUjr +∑r

(P(xj)

−1P(xr )− I)AjrUjr = 0.

The relevant eigenvector of Ajr is ujr = (1,njr )t .

• The generalization of the two-states solver writes(ujr ,P(xr − xj)

tUjr − Uj

)= 0 for all cells around the node xr ,

with unknown Ujr = (pjr ,ur ).


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Solution

pjr + σ (xr − xj ,ur )− pj + (njr ,ur − uj) = 0,∑

j ljrnjrpjr = 0.

r

jx

x

xj’’

j’

x

Vj

ddt pj +

∑r Cjr · ur = 0, Cjr = ljrnjr ,

Vjddt uj +

∑r Cjrpjr = −σ∑r Cjr ⊗ (xr − xj)ur .

- equal to the scheme in Buet, D. Franck, Numerish Mathematik, 2012.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Convergence analysis

• Performed on the problem with an additional small parameter∂tp + 1

ε∇ · u = 0,

∂tu + 1ε∇p = − σ

ε2 u,

where σ > 0 is given and ε > 0 is a small parameter.

• In the limit ε ≈ 0+, one gets asymptotically the parabolicequation

∂tp −1

σ∆p = 0.

• There is a change of type, from hyperbolic to parabolic.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Convergence

Theorem (2D on general grids) : Additionally to being WB,the scheme is AP with the error estimate

‖ph − p,uh − u‖L2([0,T ]×Ω) ≤ Ch14 ,

uniformly with respect to the small parameter ε ∈ (0, 1].

Numerical tests show convergence order between 1 and 2.

Not true for the usual edge based FV solver : this is whypropaganda is not propaganda, it is the solution for WB-AP onunstructured grids.

- Uniform convergence for a cell-centered AP discretization of the

hyperbolic heat equation on general meshes, Buet-Franck-D.-Leroy, arxiv

2014.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Section 5

Numerical results


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Meshes and P1 (E. Franck)

Kershaw mesh Random mesh

non WB-AP WB-FV edge, non AP WB-AP corner


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

2D tests

P3 is a moment based discretization of transport : n = 10S2 is a DOM discretization of transport : n = 4

idealizedP3 S2 neutron prop.

Buet, D. and Franck, Asymptotic Preserving Schemes on Distorted Meshes

for Friedrichs Systems with Stiff Relaxation : Application to Angular

Models in Linear Transport, JSC, 2014.


Introduction

dim(V) = n

dim(V) < n

dim(V) =∞

Numericalresults

Conclusions

Unification of WB based on the dual equationessentially 2 ideas : duality and linear Riemann solvers,essentially 2 families of WB solvers.

General principle : any WB-FV is a standard FV, but forα (people in the conservative word should appreciate).

Any Riemann solver can be used.High order, even FEM is possible.

Extremely nice algebra of the two-states solver for Sn.

The change-of-basis matrix P(x) seems the good object tomanipulate, particularly in multiD.

- Buet-D., The structure of well-balanced schemes for Friedrichs systems

with linear relaxation, HAL 2014


the structure of wb schemes for friedrichs systems with relaxation · 2017. 2. 21. · two-state...

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