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A Finite Volume Kinetic (FVK) framework for unsteady mixed flows in non uniform closed water pipes. Mehmet Ersoy 1, Christian Bourdarias 2 and St ephane Gerbi 3 Euskadi - Kyushu 2011 1. BCAM, Spain, mersoy@bcamath.org 2. LAMASavoie,

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SLIDE 1

A Finite Volume Kinetic (FVK) framework for unsteady mixed flows in non uniform closed water pipes.

Mehmet Ersoy 1, Christian Bourdarias 2 and St´

ephane Gerbi 3 Euskadi - Kyushu 2011

  • 1. BCAM, Spain, mersoy@bcamath.org
  • 2. LAMA–Savoie, France, christian.bourdarias@univ-savoie.fr
  • 3. LAMA–Savoie, France, stephane.gerbi@univ-savoie.fr
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SLIDE 2

Outline of the talk

Outline of the talk

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 2 / 26

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SLIDE 3

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 3 / 26

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SLIDE 4

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 4 / 26

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SLIDE 5

Unsteady mixed flows in closed water pipes ?

Free surface area (SL) sections are not completely filled and the flow is incompressible. . .

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 5 / 26

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SLIDE 6

Unsteady mixed flows in closed water pipes ?

Free surface area (SL) sections are not completely filled and the flow is incompressible. . . Pressurized area (CH) sections are non completely filled and the flow is compressible. . .

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 5 / 26

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SLIDE 7

Unsteady mixed flows in closed water pipes ?

Free surface area (SL) sections are not completely filled and the flow is incompressible. . . Pressurized area (CH) sections are non completely filled and the flow is compressible. . . Transition point

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 5 / 26

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SLIDE 8

Examples of pipes

Orange-Fish tunnel Sewers . . . in Paris Forced pipe problems . . . at Minnesota http://www.sewerhistory.org/grfx/ misc/disaster.htm

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 6 / 26

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SLIDE 9

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 7 / 26

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SLIDE 10

The PFS model

                             ∂t(A) + ∂x(Q) = 0 ∂t(Q) + ∂x Q2 A + p(x, A, E)

  • = −g A d

dxZ(x) +Pr(x, A, E) −G(x, A, E) −g K(x, S) Q|Q| A with A = Asl if E = 0 (FS state), Ach if E = 1 (P state), S = Asl if E = 0, S = |Ω(x)| if E = 1, and ...

  • C. Bourdarias, M. Ersoy and S. Gerbi

A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme.

  • Int. J. On Finite Volumes, 6(2) :1–47, 2009.
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 8 / 26

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SLIDE 11

Pressure and source terms

p = c2(A − S) + gI1(x, S) cos θ : mixed pressure law Pr = c2 Ach S − 1 d S dx + gI2(S) cos θ : pressure source term G = gAz d dx cos θ : curvature source term K = 1 K2

sRh(S)4/3

: Manning-Strickler law

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 9 / 26

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SLIDE 12

The PFS model

Mathematical properties

The PFS system is strictly hyperbolic for A(t, x) > 0. For regular solutions, the mean speed u = Q/A verifies ∂tu + ∂x u2 2 + c2 ln(A/S) + g H(S) cos θ + g Z

  • = −g K(x, S) u |u|

and for u = 0, we have : c2 ln(A/S) + g H(S) cos θ + g Z = cte where H(S) is the physical water height. There exists a mathematical entropy E(A, Q, S) = Q2 2A + c2A ln(A/S) + c2S + gz(x, S) cos θ + gAZ which satisfies ∂tE + ∂x (E u + p(x, A, E) u) = −g A K(x, S) u2 |u| 0

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 10 / 26

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SLIDE 13

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 11 / 26

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SLIDE 14

Finite Volume (VF) numerical scheme of order 1

Cell-centered numerical scheme

PFS equations under vectorial form : ∂tU(t, x) + ∂xF(x, U) = S(t, x)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 12 / 26

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SLIDE 15

Finite Volume (VF) numerical scheme of order 1

Cell-centered numerical scheme

PFS equations under vectorial form : ∂tU(t, x) + ∂xF(x, U) = S(t, x) with Un

i cte per mesh

≈ 1 ∆x

  • mi

U(tn, x) dx and S(t, x) constant per mesh,

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 12 / 26

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SLIDE 16

Finite Volume (VF) numerical scheme of order 1

Cell-centered numerical scheme

PFS equations under vectorial form : ∂tU(t, x) + ∂xF(x, U) = S(t, x) with Un

i cte per mesh

≈ 1 ∆x

  • mi

U(tn, x) dx and S(t, x) constant per mesh, Cell-centered numerical scheme : Un+1

i

= Un

i − ∆tn

∆x

  • Fi+1/2(Ui, Ui−1) − Fi−1/2(Ui−1, Ui)
  • + ∆tnS(Un

i )

where ∆tnSn

i ≈

tn+1

tn

  • mi

S(t, x) dx dt

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 12 / 26

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SLIDE 17

Finite Volume (VF) numerical scheme of order 1

Upwinded numerical scheme : the principle

PFS equations under vectorial form : ∂tU(t, x) + ∂xF(x, U) = S(t, x) with Un

i cte per mesh

≈ 1 ∆x

  • mi

U(tn, x) dx and S(t, x) constant per mesh, Upwinded numerical scheme : Un+1

i

= Un

i − ∆tn

∆x

  • Fi+1/2(Ui, Ui−1, Si,i+1) −

Fi−1/2(Ui−1, Ui, Si−1,i)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 12 / 26

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SLIDE 18

Choice of the numerical fluxes

Our goal : define Fi+1/2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A, discrete equilibrium, discrete entropy inequality

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 13 / 26

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SLIDE 19

Choice of the numerical fluxes

Our goal : define Fi+1/2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A, discrete equilibrium, discrete entropy inequality

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 13 / 26

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SLIDE 20

Choice of the numerical fluxes

Our goal : define Fi+1/2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A, discrete equilibrium, discrete entropy inequality

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 13 / 26

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SLIDE 21

Choice of the numerical fluxes

Our goal : define Fi+1/2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A, discrete equilibrium, discrete entropy inequality Our choice VFRoe solver[BEGVF] Kinetic solver[BEG10]

  • C. Bourdarias, M. Ersoy and S. Gerbi.

A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. International Journal On Finite Volumes , Vol 6(2) 1–47, 2009.

  • C. Bourdarias, M. Ersoy and S. Gerbi.

A kinetic scheme for transient mixed flows in non uniform closed pipes : a global manner to upwind all the source terms.

  • J. Sci. Comp.,pp 1-16, 10.1007/s10915-010-9456-0, 2011.
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 13 / 26

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SLIDE 22

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 14 / 26

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SLIDE 23

Principle

Density function

We introduce χ(ω) = χ(−ω) ≥ 0 ,

  • R

χ(ω)dω = 1,

  • R

ω2χ(ω)dω = 1 ,

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 15 / 26

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SLIDE 24

Principle

Gibbs Equilibrium or Maxwellian

We introduce χ(ω) = χ(−ω) ≥ 0 ,

  • R

χ(ω)dω = 1,

  • R

ω2χ(ω)dω = 1 , then we define the Gibbs equilibrium by M(t, x, ξ) = A(t, x) b(t, x) χ ξ − u(t, x) b(t, x)

  • with

b(t, x) =

  • p(t, x)

A(t, x)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 15 / 26

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SLIDE 25

Principle

micro-macroscopic relations

Since χ(ω) = χ(−ω) ≥ 0 ,

  • R

χ(ω)dω = 1,

  • R

ω2χ(ω)dω = 1 , and M(t, x, ξ) = A(t, x) b(t, x) χ ξ − u(t, x) b(t, x)

  • then

A =

  • R

M(t, x, ξ) dξ Q =

  • R

ξM(t, x, ξ) dξ Q2 A + A b2

  • p

=

  • R

ξ2M(t, x, ξ) dξ

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 15 / 26

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SLIDE 26

Principle [P02]

The kinetic formulation

(A, Q) is solution of the PFS system if and only if M satisfy the transport equation : ∂tM + ξ · ∂xM − gΦ ∂ξM = K(t, x, ξ) where K(t, x, ξ) is a collision kernel satisfying a.e. (t, x)

  • R

K dξ = 0 ,

  • R

ξ Kd ξ = 0 and Φ are the source terms.

  • B. Perthame.

Kinetic formulation of conservation laws. Oxford University Press. Oxford Lecture Series in Mathematics and its Applications, Vol 21, 2002.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 16 / 26

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SLIDE 27

Principe

The kinetic formulation

(A, Q) is solution of the PFS system if and only if M satisfy the transport equation : ∂tM + ξ · ∂xM − gΦ ∂ξM = K(t, x, ξ) where K(t, x, ξ) is a collision kernel satisfying a.e. (t, x)

  • R

K dξ = 0 ,

  • R

ξ Kd ξ = 0 and Φ are the source terms. General form of the source terms : Φ =

conservative

  • d

dxZ +

non conservative

  • B · d

dxW +

friction

K Q|Q| A2 with W = (Z, S, cos θ)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 16 / 26

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SLIDE 28

Discretization of source terms

Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting ». . .

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 17 / 26

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SLIDE 29

Discretization of source terms

Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting ». . . Then ∀(t, x) ∈ [tn, tn+1[×

  • mi

Φ(t, x) = 0 since Φ = d dxZ + B · d dxW

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 17 / 26

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SLIDE 30

Simplification of the transport equation

Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting ». . . Then ∀(t, x) ∈ [tn, tn+1[×

  • mi

Φ(t, x) = 0 since Φ = d dxZ + B · d dxW = ⇒ ∂tM + ξ · ∂xM = K(t, x, ξ)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 17 / 26

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SLIDE 31

Simplification of the transport equation

Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting ». . . Then ∀(t, x) ∈ [tn, tn+1[×

  • mi

Φ(t, x) = 0 since Φ = d dxZ + B · d dxW = ⇒    ∂tf + ξ · ∂xf = f(tn, x, ξ) = M(tn, x, ξ)

def

:= A(tn, x, ξ) b(tn, x, ξ) χ ξ − u(tn, x, ξ) b(tn, x, ξ)

  • by neglecting the collision kernel
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 17 / 26

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SLIDE 32

Discretization of source terms

On [tn, tn+1[×mi, we have :

  • ∂tf + ξ · ∂xf

= f(tn, x, ξ) = Mn

i (ξ)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 18 / 26

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SLIDE 33

Discretization of source terms

On [tn, tn+1[×mi, we have :

  • ∂tf + ξ · ∂xf

= f(tn, x, ξ) = Mn

i (ξ)

i.e. f n+1

i

(ξ) = Mn

i (ξ) + ξ ∆tn

∆x

  • M−

i+ 1

2 (ξ) − M+

i− 1

2 (ξ)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 18 / 26

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SLIDE 34

Discretization of source terms

On [tn, tn+1[×mi, we have :

  • ∂tf + ξ · ∂xf

= f(tn, x, ξ) = Mn

i (ξ)

i.e. f n+1

i

(ξ) = Mn

i (ξ) + ξ ∆tn

∆x

  • M−

i+ 1

2 (ξ) − M+

i− 1

2 (ξ)

  • where

Un+1

i

=

  • An+1

i

Qn+1

i

  • def

:=

  • R
  • 1

ξ

  • f n+1

i

(ξ) dξ

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 18 / 26

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SLIDE 35

Discretization of source terms

On [tn, tn+1[×mi, we have :

  • ∂tf + ξ · ∂xf

= f(tn, x, ξ) = Mn

i (ξ)

i.e. f n+1

i

(ξ) = Mn

i (ξ) + ξ ∆tn

∆x

  • M−

i+ 1

2 (ξ) − M+

i− 1

2 (ξ)

  • r

Un+1

i

= Un

i − ∆tn

∆x

  • F−

i+1/2 −

F+

i−1/2

  • with
  • F±

i± 1

2 =

  • R

ξ 1 ξ

  • M±

i± 1

2 (ξ) dξ.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 18 / 26

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SLIDE 36

The microscopic fluxes

Interpretation : potential bareer

M−

i+1/2(ξ) = positive transmission

  • 1{ξ>0}Mn

i (ξ)

+ 1{ξ<0, ξ2−2g∆Φn

i+1/2>0}Mn

i+1

  • −
  • ξ2 − 2g∆Φn

i+1/2

  • negative transmission
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 19 / 26

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SLIDE 37

The microscopic fluxes

Interpretation : potential bareer

M−

i+1/2(ξ) = positive transmission

  • 1{ξ>0}Mn

i (ξ)

+

reflection

  • 1{ξ<0, ξ2−2g∆Φn

i+1/2<0}Mn

i (−ξ)

+ 1{ξ<0, ξ2−2g∆Φn

i+1/2>0}Mn

i+1

  • −
  • ξ2 − 2g∆Φn

i+1/2

  • negative transmission
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 19 / 26

slide-38
SLIDE 38

The microscopic fluxes

Interpretation : potential bareer

M−

i+1/2(ξ) = positive transmission

  • 1{ξ>0}Mn

i (ξ)

+

reflection

  • 1{ξ<0, ξ2−2g∆Φn

i+1/2<0}Mn

i (−ξ)

+ 1{ξ<0, ξ2−2g∆Φn

i+1/2>0}Mn

i+1

  • −
  • ξ2 − 2g∆Φn

i+1/2

  • negative transmission

∆Φn

i+1/2 may be interpreted as a time-dependant slope !

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 19 / 26

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SLIDE 39

The microscopic fluxes

Interpretation : Dynamic slope = ⇒ Upwinding of the friction

M−

i+1/2(ξ) = positive transmission

  • 1{ξ>0}Mn

i (ξ)

+

reflection

  • 1{ξ<0, ξ2−2g∆Φn

i+1/2<0}Mn

i (−ξ)

+ 1{ξ<0, ξ2−2g∆Φn

i+1/2>0}Mn

i+1

  • −
  • ξ2 − 2g∆Φn

i+1/2

  • negative transmission

∆Φn

i+1/2 may be interpreted as a time-dependant slope !

. . . we reintegrate the friction . . .

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 19 / 26

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SLIDE 40

Upwinding of the source terms

conservative ∂xW : Wi+1 − Wi non-conservative B∂xW : B(Wi+1 − Wi) where B = 1 B(s, φ(s, Wi, Wi+1)) ds for the « straight lines paths », i.e. φ(s, Wi, Wi+1) = sWi+1 + (1 − s)Wi, s ∈ [0, 1]

  • G. Dal Maso, P. G. Lefloch and F. Murat.

Definition and weak stability of nonconservative products.

  • J. Math. Pures Appl. , Vol 74(6) 483–548, 1995.
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 20 / 26

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SLIDE 41

Numerical properties

With [ABP00] χ(ω) = 1 2 √ 31[−

√ 3, √ 3](ω)

we have : Positivity of A (under a CFL condition), Conservativity of A, Natural treatment of drying and flooding area.

for example

  • E. Audusse and M-0. Bristeau and B. Perthame.

Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report RR3989, 2000.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 21 / 26

slide-42
SLIDE 42

Numerical properties

With [ABP00] χ(ω) = 1 2 √ 31[−

√ 3, √ 3](ω)

we have : Positivity of A (under a CFL condition), Conservativity of A, Natural treatment of drying and flooding area.

for example

− → non well-balanced scheme with such a χ

  • E. Audusse and M-0. Bristeau and B. Perthame.

Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report RR3989, 2000.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 21 / 26

slide-43
SLIDE 43

Numerical properties

With [ABP00] χ(ω) = 1 2 √ 31[−

√ 3, √ 3](ω)

we have : Positivity of A (under a CFL condition), Conservativity of A, Natural treatment of drying and flooding area.

for example

− → non well-balanced scheme with such a χ − → but easy computation of the numerical fluxes

  • E. Audusse and M-0. Bristeau and B. Perthame.

Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report RR3989, 2000.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 21 / 26

slide-44
SLIDE 44

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 22 / 26

slide-45
SLIDE 45

Qualitative analysis of convergence

99 100 101 102 103 104 100 200 300 400 500 600 700 800 900 m d’eau m T = 0.000 Eau Ligne piezometrique

upstream piezometric head 104 m downstream piezometric head :

101.4 101.6 101.8 102 102.2 102.4 102.6 102.8 103 103.2 2 4 6 8 10 12 14 m d’eau Temps (s) Niveau piezometrique aval Hauteur piezo haut du tuyau
  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 23 / 26

slide-46
SLIDE 46

Convergence

During unsteady flows t = 100 s

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Erreur L2 : Ligne piezometrique au temps t = 100 s Ordre VFRoe (polyfit) = 0.91301 VFRoe (sans polyfit) Ordre FKA (polyfit) = 0.88039 FKA (sans polyfit)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 23 / 26

yL2 ln(∆x)

slide-47
SLIDE 47

Convergence

Stationary t = 500 s

  • 1.8
  • 1.6
  • 1.4
  • 1.2
  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Erreur L2 : Ligne piezometrique au temps t = 500 s Ordre VFRoe (polyfit) = 1.0742 VFRoe (sans polyfit) Ordre FKA (polyfit) = 1.0371 FKA (sans polyfit)

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 23 / 26

yL2 ln(∆x)

slide-48
SLIDE 48

Outline

Outline

1 Unsteady mixed flows : PFS equations (Pressurized

and Free Surface)

Definitions and examples The PFS model

2 A FVK Framework

Kinetic Formulation and numerical scheme Numerical results

3 Conclusion and perspectives

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 24 / 26

slide-49
SLIDE 49

Conclusion

Easy implementation even if source terms are complex. Very good agreement for uniform pipe.

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 25 / 26

slide-50
SLIDE 50

Conclusion and perspectives

Easy implementation even if source terms are complex. Very good agreement for uniform pipe. To do : Upwinding scheme = ⇒ a priori preservation of steady states (in progress). Discrete entropy inequalities ?

  • M. Ersoy (BCAM)

FVK framework for PFS-model Euskadi - Kyushu 2011 25 / 26

slide-51
SLIDE 51

Thank you

Thank you

for your

for your

attention

attention

4 × 4 9 Mars 2011

  • M. Ersoy (BCAM)

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