A well-balanced and asymptotic-preserving scheme for a 1-D diesel particulate filters model with dominant source and porosity effects Quang Huy T RAN joint work with Frédéric C OQUEL

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

1

1 Physical context and motivation

Outline 1

Physical context and motivation Full DPF model Simplified DPF model Two special cases

2

Toward a new scheme for the subsonic DPF

3

Numerical results

4

Conclusion

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

2

1 Physical context and motivation

1.1 DPF model

Diesel Particulate Filter Trapped PM

PM CO HCs PAHs SO2 NO

Plugged channels CO2 H2 0 SO2 NO

Remove particulate matter from the exhaust gas of a diesel engine by forcing it to flow through the solid walls between the channels of a complex honeycomb structure. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

3

1 Physical context and motivation

1.1 DPF model

DPF model For the gas

∂t (φ ) = 0, ∂t (φ ρ ) + ∇· (φ ρ u) = 0, ∂t (φ ρ u) + ∇· (φ ρ u ⊗ u) + φ ∇ p = − φ h(φ , u), ∂t (φ ρ e) + ∇· (φ ρ eu + φ pu) = − φ h(φ , u) · u + ∇· (λ ∇ T) + ησ (Tσ − T), with

e=

u2 + ε, 2

ε = cv T,

p = (γ − 1)ρε .

(1a) (1b) (1c) (1d)

(2)

For the solid

∂t ((1 − φ )ρσ εσ ) = ∇· (λσ ∇ Tσ ) + ησ (T − Tσ ), with NAHSSTA’09 (Castro-Urdiales)

ρσ = ρσ0 ,

εσ = cv,σ Tσ .

WB-AP scheme for DPF model

(3) (4) 08/09/2009

4

1 Physical context and motivation

1.1 DPF model

DPF model

Darcy-Forchheimer drag forces   1−φ 2 µ 1−φ h(φ , u) = β0 ρ kuku. u+ φ k0 φ

(5)

combined with Carman-Kozeny’s relation. Other applications using the same model fast gas flow across solid grids (2-D) [Rochette & Clain, J. Comput. Phys. 219, 2006]. internal arc fault in medium voltage cell (1-D) [Rochette, Clain & Gentils, IEEE Trans. Power Delivery 23, 2008].

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

5

1 Physical context and motivation

1.1 DPF model

Typical features Engine

φ =1

Filter φ 0 and 2pρ + ρ pρρ > 0.

(9)

subject to Source terms K (φ , u) = κ (φ , B|u|)Bu,

B ≫ 1,

(10)

where κ (φ , |v|) is a nonzero polynomial in |v| with nonnegative coefficients for every fixed φ . NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

8

1 Physical context and motivation

1.2 Simplified DPF model

Simplified DPF model Properties of source terms The value κ (φ , |v|) is positive except at |v| = 0; The mapping v ∈ R 7→ κ (φ , |v|)v ∈ R is strictly increasing and invertible;

The Darcy-Forchheimer example   1−φ 2 1−φ K(φ , u) = F|u|u. Du + φ φ D √ = F, with d0 constant, and where B = d0   1−φ 1−φ 2 d0 + κ (φ , |v|) = |v|, φ φ satisfies the assumptions for φ < 1.

NAHSSTA’09 (Castro-Urdiales)

(11)

(12)



WB-AP scheme for DPF model

08/09/2009

9

1 Physical context and motivation

1.2 Simplified DPF model

Simplified DPF model p Let c = pρ (ρ ). For u 6= c, system (7) is hyperbolic. Its eigenvalues are given by u − c, 0, u + c. (13) Loss of the eigenvectors basis for u = ±c.



The characteristic speed 0 is associated with the porosity (jump in φ ) and is linearly degenerate The characteristic speeds u ± c are associated with acoustic waves and are genuinely nonlinear, due to the assumptions on p.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

10

1 Physical context and motivation

1.3 Two special cases

Nozzle flow Putting B = 0, we have

∂t (φ ) ∂t (φ ρ ) + ∂x (φ ρ u)

= 0,

(14a)

= 0,

(14b)

∂t (φ ρ u) + ∂x (φ ρ u ) + φ ∂x p = 0.

(14c)

2

Sense of non-conservative product and understanding of resonance. Theoretical works Notion of weak solution [Dal Maso, LeFloch & Murat, 1995] Analysis of Riemann problem [Andrianov & Warnecke, 2004], [LeFloch & Thanh, 2003]

Numerical schemes Hydrostatic reconstruction [Bouchut, 2004] Standing wave reconstruction [Kröner & Thanh, 2006]

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

11

1 Physical context and motivation

1.3 Two special cases

Euler with friction Putting φ = Cte < 1, we have

∂t (ρ ) + ∂x (ρ u)

= 0,

(15a)

2

∂t (ρ u) + ∂x (ρ u ) + ∂x p = − ρ K (u).

(15b)

Preservation of steady states and consistency with porous media limit. Theoretical works Existence and uniqueness [Perthame, Souganidis & Lions, 1996] Darcy parabolic limit [Marcati & Milani, 1990]

Numerical schemes Well-balanced scheme using non-conservative reformulation [Cargo & Leroux , 1994], [Greenberg & Leroux, 1996] Asymptotic-preserving USI scheme [Bouchut, Ounaissa & Perthame, 2007]

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

12

3 Toward a new scheme for the subsonic DPF

Outline 1

Physical context and motivation

2

Toward a new scheme for the subsonic DPF Two convenient tools Riemann solver WB and AP properties

3

Numerical results

4

Conclusion

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

13

3 Toward a new scheme for the subsonic DPF

3.1 Two convenient tools

Cargo-Leroux’s procedure Enlarged system

∂t (φ ) ∂t (φ ρ ) + ∂x (φ ρ u)

= 0,

(16a)

= 0,

(16b)

2

∂t (φ ρ u) + ∂x (φ ρ u ) + φ ∂x p = − K(φ , u)∂x m, ∂t (φ ρ m) + ∂x (φ ρ mu) = 0.

(16c) (16d)

By virtue of (16b), there exists a function m (t, x) such that

∂t m = −φ ρ u

and

∂x m = φ ρ .

(17)

Think of m as an independent unknown satisfying ∂t m + u∂x m = 0, which yields (16d). LHS and RHS are now at the same “level” and can be dealt with simultaneously. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

14

3 Toward a new scheme for the subsonic DPF

3.1 Two convenient tools

Properties of the enlarged system

For u 6= ±c, system (16) is hyperbolic. Its eigenvalues are given by u − c, 0, u, u + c. Loss of the eigenvectors basis for u = ±c.

(18)



The characteristic speed 0 is associated with the porosity (jump in φ ), whereas the characteristic speed u is associated with the friction (jump in m). These two eigenfields are linearly degenerate. The characteristic speeds u ± c are associated with acoustic waves and are genuinely nonlinear, due to the assumptions on p.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

15

3 Toward a new scheme for the subsonic DPF

3.1 Two convenient tools

Relaxation procedure Relaxation system

∂t (φ ) ∂t (φ ρ ) + ∂x (φ ρ u)

= 0,

(19a)

= 0,

(19b)

∂t (φ ρ u) + ∂x (φ ρ u2 ) + φ ∂x Π = − K(φ , u)∂x m, ∂t (φ ρ T ) + ∂x (φ ρ Tu) = λ φ ρ (τ − T), ∂t (φ ρ m) + ∂x (φ ρ mu) = 0.

(19c) (19d) (19e)

Relaxation pressure T) = p(T) + a2 (T − τ ) Π (τ ,T

(20)

where τ = 1/ρ is the specific volume and T is the relaxation counterpart of τ . For λ → +∞, we have T → τ .

a > 0 must be large enough [Coquel et al., 1999], [Bouchut, 2002]. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

16

3 Toward a new scheme for the subsonic DPF

3.1 Two convenient tools

Properties of the relaxation DPF model For a > 0 and u 6= ±aτ , system (19) is hyperbolic. Its eigenvalues are all linearly degenerate and given by u − aτ , 0, u, u, u + aτ . Loss of the eigenvectors basis for u = ±aτ .

(21)



Thanks to linear degeneracy, the Riemann invariants u − aτ o u u + aτ

φ , m, T, u − aτ 2

(22a) 2 2

m, T, φ ρ u, u − a τ

(22b)

φ , u, φ Π + Km

(22c)

φ , m, T, u + aτ

(22d)

might make the Riemann problem easier to solve “by hands” in order to apply the Godunov method. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

17

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Solving the Riemann problem ?

FAQ (Frequently Annoying Questions) Assume an ordering for the waves. Write down the algebraic relations between intermediate states. There may be no solutions at all or no solutions within our skills. . . There may be one or several solutions, but to check that these are consistent with the ordering assumption may be a difficult task.

Claims for the present case In the subsonic regime, we only need to consider two cases for the ordering of waves. It is possible to completely solve each case and to check consistency. It is possible to predict in which one we are in an a priori manner.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

18

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Solving the Riemann problem ?

FAQ (Frequently Annoying Questions) Assume an ordering for the waves. Write down the algebraic relations between intermediate states. There may be no solutions at all or no solutions within our skills. . . There may be one or several solutions, but to check that these are consistent with the ordering assumption may be a difficult task.

Claims for the present case In the subsonic regime, we only need to consider two cases for the ordering of waves. It is possible to completely solve each case and to check consistency. It is possible to predict in which one we are in an a priori manner.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

18

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Case 1

t o

u − aτ

φR , τL♯ , u♯ TL , mL

u

φR , τR♯ , u♯ TR , mR

φL , τ ♭ , u♭ TL , mL

u + aτ

φL , τL , uL , TL , mL

φR , τR , uR , TR , mR x

uL − aτL = u♭ − aτ ♭

u − aτ o

u u + aτ

φL ρ ♭ u♭ = φR ρL♯ u♯

(23b)

(u♭ )2 − a2 (τ ♭ )2 = (u♯ )2 − a2 (τL♯ )2

(23c)

=

(23d)

φR Π(τL♯ , TL ) + K(φR , u♯ )mL u♯ + aτ2♯

NAHSSTA’09 (Castro-Urdiales)

(23a)

φR Π(τR♯ , TR ) + K(φR , u♯ )mR

= uR + aτR

WB-AP scheme for DPF model

(23e) 08/09/2009

19

3 Toward a new scheme for the subsonic DPF

Case 2

3.2 Riemann solver

t

♯ ♯ u φ L , τR , u o TR , mR φL , τL♯ , u♯ TL , mL

φR , τ ♭ , u♭ TR , mR

u + aτ

u − aτ

φR , τR , uR , TR , mR

φL , τL , uL , TL , mL

x

u − aτ u

uL − aτL = u♯ − aτL♯

(24a)

φL Π(τL♯ , TL ) + K(φL , u♯ )mL = φL Π(τR♯ , TR ) + K(φL , u♯ )mR (24b) φL ρR♯ u♯ = φR ρ ♭ u♭

o ♯ 2

(u ) u + aτ NAHSSTA’09 (Castro-Urdiales)

− a (τR♯ )2 ♭ ♭ 2

♭ 2

(24c) 2

♭ 2

= (u ) − a (τ )

u + aτ = uR + aτR

WB-AP scheme for DPF model

(24d) (24e) 08/09/2009

20

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Yes we can (predict) ! Theorem Assume that the left and right states are subsonic. Let u∗ =

uL + uR ΠR − ΠL − . 2 2a

(25)

sgn u♯ = sgn u∗ .

(26)

Then, In other words, we are in Case 1 if u∗ > 0; otherwise, we are in Case 2. Furthermore, when φ = 1 and B = 0 (no porosity, no source), u♯ = u∗ .

(27)

The intermediate velocity u∗ of the isentropic model (no porosity, no source) plays a fundamental role in the DPF model (with porosity, with source). NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

21

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Yes we can (solve) ! Λ=

φR , φL

∆m = mR − mL =

(φ ρ )L + (φ ρ )R ∆x. 2

(28)

Theorem Assume subsonic states and u∗ > 0. Then, the intermediate Mach number µL♯ = u♯ /aτL♯ is the unique solution in [0, min{1, Λ−1 }[ of the equation 

where u♯ =

1 − ΛµL♯

(1 − µL♯ )(1 + µL♯ )

   = 1 + ΛµL♯ 1 + 1 + a−1 Bκ (φR , B|uu♯ |)∆m µL♯ 2 aτL (1 − µL )µL♯ q 1 − (µL♯ )2

q 1 + ΛµL♯ q , ♯ 1 − ΛµL

µL∗ =

u∗ , aτL∗



1 − µL∗ 1 + µL∗

τL∗ = τL +

2

,

(29)

u∗ − uL . (30) a

Furthermore, when φ = 1 and B = 0, we have µL♯ = µL∗ . NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

22

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Yes we can (solve more quickly) ! In practice, we recommend to “freeze” κ (φR , B|uu♯ |) at κ (φR , B|u˜ |), where u˜ is the unique root of u˜ +

∆m κ (φR , B|˜u|)B˜u = u∗ . 2aφR

(31)

This reduces equation (29) to a third-degree polynomial in µL♯ . The replacement value (31) has been designed so as to maintain the asymptotic-preserving property of the scheme. When Λ = 1, we have u♯ = u˜ . More about u˜ in [Chalons et al., 2009] and in Edwige G ODLEWSKI’s talk (tomorrow afternoon) on the Euler system with gravity and friction.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

23

3 Toward a new scheme for the subsonic DPF

3.2 Riemann solver

Update formulae (subsonic regime) (φ ρ u)i+1/2 − (φ ρ u)i−1/2 (φ ρb)i − (φ ρ )i + = 0 ∆t ∆x 2 L 2 R u)i − (φ ρ u)i (φ ρ u + φ Π)i+1/2 − (φ ρ u + φ Π)i−1/2 (φ ρbb + ∆t ∆x ∆mi+1/2 ∆mi−1/2 K(φi , (u♯i−1/2 )+ ) − K(φi , (u♯i+1/2 )− ) =− ∆x ∆x

Theorem If a is large enough (computable lower-bound) and if 1 ∆t max max{|vi − aτi |, |v + aτi |} < , ∆x i∈Z 2

(33)

then NAHSSTA’09 (Castro-Urdiales)

ρbi > 0.

WB-AP scheme for DPF model

(34) 08/09/2009

24

3 Toward a new scheme for the subsonic DPF

3.3 Properties

Well-balanced behavior Proposition The proposed scheme, with either the exact equation (29) or the simplified version (31), preserves the discrete steady states at rest

φ 6= Cte,

ρ = Cte,

u = 0.

(35)

This property is not trivial because of φ 6= Cte.

In the DPF problem, the steady states of interest are in fact

φ 6= Cte, φ ρ u = q0 (6= 0), q0 dx u + φ dx p = −q0 κ (φ , B|u|)B.

(36a) (36b) (36c)

It is numerically observed that these are well preserved. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

25

3 Toward a new scheme for the subsonic DPF

3.3 Properties

Asymptotic behavior For B → +∞, we postulate the asymptotic expansions 1 ρ = ρ 0+ ρ 1 + . . . B 1 u = u0 + u1 + . . . B

(37a) (37b)

Inserting this into the momentum balance and looking at terms in ◦ O(B1+d κ ), we have u0 ≡ 0.

(38)

In order to analyze the long-time behavior of the model and of the scheme, we perform the rescaling s=

NAHSSTA’09 (Castro-Urdiales)

t B

and

v = Bu.

WB-AP scheme for DPF model

(39)

08/09/2009

26

3 Toward a new scheme for the subsonic DPF

3.3 Properties

Asymptotic-preserving behavior Theorem The proposed scheme, with either the exact equation (29) or the simplified version (31), is asymptotic-preserving: when B → +∞, the limit scheme (at fixed ∆x) behaves as a consistent discretization of the limit model. The limit model is a porous media model

∂s (φ ) = 0, ∂s (φ ρ ) + ∂x (φ ρ v) = 0, ∂x (p) = − ρκ (φ , |vv|)vv.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

(40a) (40b) (40c)

08/09/2009

27

3 Toward a new scheme for the subsonic DPF

3.3 Properties

Asymptotic-preserving behavior The limit scheme reads (φ ρ )i+1/2 vi+1/2 − (φ ρ )i−1/2 vi−1/2 (φ ρb)i − (φ ρ )i + =0 ∆s ∆x φi+1/2 ∆x pi+1 − pi = κ (φi+1/2 , |vvi+1/2 |)vvi+1/2 − ∆mi+1/2 ∆x with 

φi+1/2 =

φi 1{vi+1/2 0} + φi ρi+1 1{vi+1/2 0} +

WB-AP scheme for DPF model

08/09/2009

28

4 Results

Outline

1

Physical context and motivation

2

Toward a new scheme for the subsonic DPF

3

Numerical results

4

Conclusion

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

29

4 Results

Typical test case Domain length L = 0.6 m. Steady state corresponding to a discontinuous transition in porosity

φ (x) = 1 × 1{x0.3} ;

(43)

large friction coefficients D = 3 × 104

and

F = 3 × 106 ;

(44)

the inlet flow-rate condition

φ ρ u(t, x = 0) = 20 kg/m2 s;

(45)

the outlet pressure condition p(t, x = 0.6) = 105 Pa

(46)

and L/∆x = 50, 100, 200, 400. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

30

4 Results

Splitting scheme 1.6

Statio. 50 100 200 400

P (Bar)

1.4

4 3.5 u (m/s)

1.5

1.3

3 2.5

1.2 2 1.1 1.5 1 0

0.1

0.2

0.3 x

0.4

0.5

0.6

0.1

0.2

0.3 x

0.4

0.5

0.6

0

0.1

0.2

0.3 x

0.4

0.5

0.6

2.75 u^2/2 + h (m2/ s2/ 10^4)

Phi* rho* u (kg/ m2/ s)

20.5

0

20 19.5 19 18.5 18 17.5 17

2.7 2.65 2.6 2.55 2.5 2.45 2.4 2.35 2.3

0

0.1

0.2

0.3 x

NAHSSTA’09 (Castro-Urdiales)

0.4

0.5

0.6

WB-AP scheme for DPF model

08/09/2009

31

4 Results

Relaxation scheme 1.6

Statio. 50 100 200 400

P (Bar)

1.4

4 3.5 u (m/s)

1.5

1.3

3 2.5

1.2 2 1.1 1.5 1 0

0.1

0.2

0.3 x

0.4

0.5

0.6

0.1

0.2

0.3 x

0.4

0.5

0.6

0

0.1

0.2

0.3 x

0.4

0.5

0.6

2.75 u^2/2 + h (m2/ s2/ 10^4)

Phi* rho* u (kg/ m2/ s)

20.5

0

20 19.5 19 18.5 18 17.5 17

2.7 2.65 2.6 2.55 2.5 2.45 2.4 2.35 2.3

0

0.1

0.2

0.3 x

NAHSSTA’09 (Castro-Urdiales)

0.4

0.5

0.6

WB-AP scheme for DPF model

08/09/2009

32

4 Results

Relaxation scheme (close-up) 1.6

Statio. 50 100 200 400

P (Bar)

1.4

4 3.5 u (m/s)

1.5

1.3

3 2.5

1.2 2 1.1 1.5 1 0

0.1

0.2

0.3 x

0.4

0.5

0.6

0.1

0.2

0.3 x

0.4

0.5

0.6

0

0.1

0.2

0.3 x

0.4

0.5

0.6

2.75 u^2/2 + h (m2/ s2/ 10^4)

Phi* rho* u (kg/ m2/ s)

20.05

0

20 19.95 19.9 19.85 19.8 19.75

2.7 2.65 2.6 2.55 2.5 2.45 2.4 2.35 2.3

0

0.1

0.2

0.3 x

NAHSSTA’09 (Castro-Urdiales)

0.4

0.5

0.6

WB-AP scheme for DPF model

08/09/2009

33

Conclusion

Outline

1

Physical context and motivation

2

Toward a new scheme for the subsonic DPF

3

Numerical results

4

Conclusion

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

34

Conclusion

Conclusion

Present contribution Positive, well-balanced and asymptotic-preserving scheme based on relaxation for a 1-D Euler-like system with friction and porosity under subsonic regimes. Cargo-Leroux’s procedure and the relaxation philosophy have been of great help for our purpose.

Future investigations for the simplified DPF model Discrete energy inequality satisfied by the scheme ? Preservation of “subsonicity” ? From subsonic to supersonic : how to get across resonance ?

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

35

Various approaches

Apply schemes with centered source terms (e.g., splitting) Computationally very expensive. Steady states not preservedpand asymptotic behavior only if the mesh size is smaller than 1/ β0 .

Couple a hyperbolic domain with a parabolic domain Interesting for engineers since old codes can be re-used, but lack of rigorous treatments to capture transition. Work out a new hyperbolic scheme This new scheme should be capable of dealing with the two extreme regimes and have desirable properties by construction.

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

36

Energy (in)equalities Smooth solutions of the simplified DPF model (7) satisfy the additional energy balance law

∂t (φ ρ e) + ∂x (φ ρ eu + φ pu) = −φ ρκ (φ , B|u|)B|u|2 ,

(47)

where

p u2 ερ = 2. + ε, (48) 2 ρ A non-smooth solution is said to be entropic if “=” is replaced by “≤”. Solutions of the relaxation DPF model (19) satisfy the additional energy balance law e=

∂t (φ ρ E ) + ∂x (φ ρ E u + φ Πu) = − κ (φ , B|u|)B|u|2 ∂x m

(49)

− λ φ ρ [a2 + pτ (T)](τ − T)2 ,

where E = NAHSSTA’09 (Castro-Urdiales)

u2 Π2 (τ , T) − p2 (T) . + ε (T) + 2 2a2 WB-AP scheme for DPF model

(50) 08/09/2009

37

Subcharacteristic condition Proposition The relaxation system (19) is a “good” approximation of the original system (7) for λ → +∞ under the Whitham condition p a > −pτ (τ ) (51) for all τ in the range of the problem under consideration. By a linear analysis via the Chapman-Enskog expansion, we show that

∂t (φ ρ u) + ∂x (φ ρ u2 ) + φ ∂x p = − φ ρ K(φ , u)  2  a + pτ (τ ) φ ∂x (φ u) . + ∂x λ φρ

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

(52)

08/09/2009

38

Explicit time-integration Two-step procedure 

n φ  φρ     φ ρu    φ ρT  φ ρm x  

Tn := τZn mn :=

x

(φ ρ )n



n+1 φ  φρ    λ =0  −−−−→   φ ρu  φ ρT  φ ρm   y free

−−−−→ evolution

T 6= τZ

m 6=

x

φρ

λ =+∞

−−−−→

return to

−−−−−−→ equilibrium



n+1 φ  φρ     φ ρu    φ ρT  φ ρm x  

Tn+1 := τZn+1

mn+1 :=

x

(φ ρ )n+1

For the free evolution step (λ = 0), use the Godunov scheme. This requires that we know the solution of the Riemann problem. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

39

Details for Case 1 From µL♯ , we infer

τ♭ =

τL (1 − µL )

u♭ =

aτL (1 − µL )ΛµL♯

1 − ΛµL♯

1 − ΛµL♯

q ♯ τL (1 − µL ) 1 + ΛµL ♯ q τL = q 1 − (µL♯ )2 1 − ΛµL♯ q ♯ ♯ aτL (1 − µL )µL 1 + ΛµL ♯ q u = q 1 − (µL♯ )2 1 − ΛµL♯

(53a)

(53b)

To guarantee that

τR♯ = τR +

uR − u♯ a

(54)

remains positive, use the estimate

µL♯ ≤

2µL∗ 1+Λ

(55)

to derive the sufficient condition (58). NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

40

Conditions on a Subcharacteristic condition a>

p

−pτ (τ )

(56)

for all τ in the range of the problem under consideration. Positivity of intermediate densities for uniform porosity min{τL , τR }a2 +

|ΠR − ΠL | uR − uL a− > 0. 2 2

(57)

Positivity of intermediate densities for variable porosity min{τL , τR }a2 +

NAHSSTA’09 (Castro-Urdiales)

min{φL , φR }|ΠR − ΠL | φR uR − φL uL a+ > 0, φL + φR φL + φR

WB-AP scheme for DPF model

(58)

08/09/2009

41

Asymptotic behavior In the limit B → +∞, the update equation for the momentum balance becomes pi+1/2 − pi−1/2 ∆mi−1/2 κ (φi−1/2 , |vi−1/2 |)vi−1/2 =− ∆x 2φi−1/2 ∆x ∆mi+1/2 − κ (φi+1/2 , |vi+1/2 |)vi+1/2 2φi+1/2 ∆x

(59)

where

pi + pi+1 . 2 This can also be deduced from the definition of vi+1/2 . pi+1/2 =

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

(60)

08/09/2009

42

Euler isentropic model revisited Euler isentropic model

∂t (ρ ) + ∂x (ρ u)

= 0,

(61a)

∂t (ρ u) + ∂x (ρ u + p ) = 0.

(61b)

2

where p = p(ρ ) is subject to conditions (9). Relaxation model

∂t (ρ ) + ∂x (ρ u)

=0,

(62a)

2

∂t (ρ u) + ∂x (ρ u + Π ) = 0, ∂t (ρ T) + ∂x (ρ Tu) = λ ρ (τ − T),

(62b) (62c)

where T is the relaxation counterpart of τ = 1/ρ for λ → ∞, and T) = p(T) + a2 (T − τ ) Π (τ ,T

(63)

is the relaxation pressure, defined for a > 0 large enough. NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

43

Properties of the relaxation isentropic model For a > 0, system (62) is hyperbolic. Its eigenvalues are all linearly degenerate and given by u − aτ , u, u + aτ .

(64)

To be compared with u ± c for the isentropic model.

Thanks to linear degeneracy, the Riemann invariants u − aτ

T, u − aτ u, Π

(65b)

u + aτ

T, u + aτ

(65c)

u

(65a)

enable us to solve the Riemann problem “by hands.”

NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

08/09/2009

44

Riemann problem

t u

τL∗ , u∗ , TL

τR∗ , u∗ , TR

u − aτ

u + aτ

τL , uL , TL

τR , uR , TR x



uL + uR ΠR − Π L − 2 2a uR − uL ΠR − Π L ∗ − τL − τL = 2a 2a2 uR − uL ΠR − Π L + τR∗ − τR = 2a 2a2 ΠL + ΠR uR − uL Π∗L = Π∗R = −a 2 2

NAHSSTA’09 (Castro-Urdiales)

u∗ =

WB-AP scheme for DPF model

(66a) (66b) (66c) (66d) 08/09/2009

45

A technical requirement Lemma Assume the relaxation parameter a is large enough, that is, min{τL , τR }a2 +

|ΠR − ΠL | uR − uL a− > 0. 2 2

(67)

The intermediate states have positive densities, i.e., τL∗ > 0 and τR∗ > 0; We have the ordering uL − aτL < u∗ < uR + aτR ;

The “subsonicity” of the left and right states uL uR µL := µR := ∈ ] − 1, 1[ and ∈ ] − 1, 1[ aτL aτR implies that of the upwinded intermediate state, namely, u∗ µL∗ := ∗ ∈ ]0, 1[ if u∗ > 0, aτL u∗ µR∗ := ∗ ∈ ] − 1, 0[ if u∗ < 0. aτR NAHSSTA’09 (Castro-Urdiales)

WB-AP scheme for DPF model

(68)

(69a) (69b) 08/09/2009

46

Update formulae (subsonic regime)

where

(ρ u)∗i+1/2 − (ρ u)∗i−1/2 (ρb)i − (ρ )i + = 0, ∆t ∆x 2 ∗ 2 ∗ (ρbb u)i − (ρ u)i (ρ u + Π)i+1/2 − (ρ u + Π)i−1/2 + = 0, ∆t ∆x (ρ u)∗ = ρL∗ (u∗ )+ + ρR∗ (u∗ )− .

(70a) (70b)

(71)

Theorem If a is large enough and if 1 ∆t max max{|vi − aτi |, |v + aτi |} < , ∆x i∈Z 2

(72)

then NAHSSTA’09 (Castro-Urdiales)

ρbi > 0.

WB-AP scheme for DPF model

(73) 08/09/2009

47

Summary of various conditions on a Subcharacteristic condition a>

p

−pτ (τ )

(74)

for all τ in the range of the problem under consideration. Positivity of intermediate densities for uniform porosity min{τL , τR }a2 +

|ΠR − ΠL | uR − uL a− > 0. 2 2

(75)

Positivity of intermediate densities for variable porosity min{τL , τR }a2 +

NAHSSTA’09 (Castro-Urdiales)

min{φL , φR }|ΠR − ΠL | φR uR − φL uL a+ > 0. φL + φR φL + φR

WB-AP scheme for DPF model

(76)

08/09/2009

48