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A Well Balanced Finite Volume Kinetic (FVK) scheme for unsteady mixed flows in non uniform closed water pipes. Mehmet Ersoy 1 , Christian Bourdarias 2 and St ephane Gerbi 3 IMATH, November 24, 2011 1. IMATH, Toulon, Mehmet.Ersoy@univ-tln.fr

  1. The free surface model  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − gA sl dx + Pr sl ( x, A sl ) − G ( x, A sl )  A sl        with p sl = gI 1 ( x, A sl ) cos θ : hydrostatic pressure law Pr sl = gI 2 ( x, A sl ) cos θ : pressure source term gA sl z d G = dx cos θ : curvature source term M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 13 / 50

  2. The free surface model  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − gA sl dx + Pr sl ( x, A sl ) − G ( x, A sl )  A sl gK ( x, A sl ) Q sl | Q sl |   −    A sl   � �� � friction added after the derivation with p sl = gI 1 ( x, A sl ) cos θ : hydrostatic pressure law Pr sl = gI 2 ( x, A sl ) cos θ : pressure source term gA sl z d G = dx cos θ : curvature source term 1 K = : Manning-Strickler law K 2 s R h ( A sl ) 4 / 3 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 13 / 50

  3. Derivation of the pressurized model 3 D isentropic compressible equations ∂ t ρ + div( ρ U ) = 0 ∂ t ( ρ U ) + div( ρ U ⊗ U ) + ∇ p = ρ F with p = p a + ρ − ρ 0 with c sound speed c 2 Method : 1 Write Euler equations in curvilinear coordinates. 2 Write equations in non-dimensional form using the small parameter ǫ = H/L and takes ǫ = 0 . 3 Section averaging ρU ≈ ρU and ρU 2 ≈ ρU U . 4 Introduce A ch ( t, x ) : equivalent wet area, Q ch ( t, x ) discharge given by : A ch ( t, x ) = ρ S ( x ) , Q ch ( t, x ) = A ch ( t, x ) u ( t, x ) ρ 0 � 1 u ( t, x ) = U ( t, x ) dydz S ( x ) Ω( x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 14 / 50

  4. Derivation of the pressurized model 3 D isentropic compressible equations ∂ t ρ + div( ρ U ) = 0 ∂ t ( ρ U ) + div( ρ U ⊗ U ) + ∇ p = ρ F with p = p a + ρ − ρ 0 with c sound speed c 2 Method : 1 Write Euler equations in curvilinear coordinates. 2 Write equations in non-dimensional form using the small parameter ǫ = H/L and takes ǫ = 0 . 3 Section averaging ρU ≈ ρU and ρU 2 ≈ ρU U . 4 Introduce A ch ( t, x ) : equivalent wet area, Q ch ( t, x ) discharge given by : A ch ( t, x ) = ρ S ( x ) , Q ch ( t, x ) = A ch ( t, x ) u ( t, x ) ρ 0 � 1 u ( t, x ) = U ( t, x ) dydz S ( x ) Ω( x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 14 / 50

  5. Derivation of the pressurized model 3 D isentropic compressible equations ∂ t ρ + div( ρ U ) = 0 ∂ t ( ρ U ) + div( ρ U ⊗ U ) + ∇ p = ρ F with p = p a + ρ − ρ 0 with c sound speed c 2 Method : 1 Write Euler equations in curvilinear coordinates. 2 Write equations in non-dimensional form using the small parameter ǫ = H/L and takes ǫ = 0 . 3 Section averaging ρU ≈ ρU and ρU 2 ≈ ρU U . 4 Introduce A ch ( t, x ) : equivalent wet area, Q ch ( t, x ) discharge given by : A ch ( t, x ) = ρ S ( x ) , Q ch ( t, x ) = A ch ( t, x ) u ( t, x ) ρ 0 � 1 u ( t, x ) = U ( t, x ) dydz S ( x ) Ω( x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 14 / 50

  6. Derivation of the pressurized model 3 D isentropic compressible equations ∂ t ρ + div( ρ U ) = 0 ∂ t ( ρ U ) + div( ρ U ⊗ U ) + ∇ p = ρ F with p = p a + ρ − ρ 0 with c sound speed c 2 Method : 1 Write Euler equations in curvilinear coordinates. 2 Write equations in non-dimensional form using the small parameter ǫ = H/L and takes ǫ = 0 . 3 Section averaging ρU ≈ ρU and ρU 2 ≈ ρU U . 4 Introduce A ch ( t, x ) : equivalent wet area, Q ch ( t, x ) discharge given by : A ch ( t, x ) = ρ S ( x ) , Q ch ( t, x ) = A ch ( t, x ) u ( t, x ) ρ 0 � 1 u ( t, x ) = U ( t, x ) dydz S ( x ) Ω( x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 14 / 50

  7. The pressurized model  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − gA ch dx + Pr ch ( x, A ch ) − G ( x, A ch )  A ch        with c 2 ( A ch − S ) p ch = : acoustic type pressure law � A ch � d S c 2 Pr ch = − 1 : pressure source term S dx gA ch z d G = dx cos θ : curvature source term M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 15 / 50

  8. The pressurized model  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − gA ch dx + Pr ch ( x, A ch ) − G ( x, A ch )  A ch gK ( x, S ) Q ch | Q ch |   −    A ch   � �� � friction added after the derivation with c 2 ( A ch − S ) p ch = : acoustic type pressure law � A ch � d S c 2 Pr ch = − 1 : pressure source term S dx gA ch z d G = dx cos θ : curvature source term 1 K = : Manning-Strickler law K 2 s R h ( S ) 4 / 3 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 15 / 50

  9. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 16 / 50

  10. the PFS model Models are formally close . . .  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − g A sl dx + Pr sl ( x, A sl )  A sl  − G ( x, A sl )     − gK ( x, A sl ) Q sl | Q sl |   A sl  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − g A ch dx + Pr ch ( x, A ch )  A ch  − G ( x, A ch )     − gK ( x, S ) Q ch | Q ch |   A ch M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 17 / 50

  11. the PFS model Models are formally close . . .  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − g A sl dx + Pr sl ( x, A sl )  A sl  − G ( x, A sl )     − gK ( x, A sl ) Q sl | Q sl |   A sl  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − g A ch dx + Pr ch ( x, A ch )  A ch  − G ( x, A ch )     − gK ( x, S ) Q ch | Q ch |   A ch Continuity criterion M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 17 / 50

  12. the PFS model Models are formally close . . .  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − g A sl dx + Pr sl ( x, A sl )  A sl  − G ( x, A sl )     − gK ( x, A sl ) Q sl | Q sl |   A sl  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − g A ch dx + Pr ch ( x, A ch )  A ch  − G ( x, A ch )     − gK ( x, S ) Q ch | Q ch |   A ch « mixed » condition M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 17 / 50

  13. the PFS model Models are formally close . . .  ∂ t A sl + ∂ x Q sl = 0 ,   � Q 2 �   d Z   sl ∂ t Q sl + ∂ x + p sl ( x, A sl ) = − g A sl dx + Pr sl ( x, A sl )  A sl  − G ( x, A sl )     − gK ( x, A sl ) Q sl | Q sl |   A sl  ∂ t A ch + ∂ x Q ch = 0 ,   � Q 2 �   d Z   ch ∂ t Q ch + ∂ x + p ch ( x, A ch ) = − g A ch dx + Pr ch ( x, A ch )  A ch  − G ( x, A ch )     − gK ( x, S ) Q ch | Q ch |   A ch To be coupled M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 17 / 50

  14. The PFS model The « mixed » variable We introduce a state indicator � 1 if the flow is pressurized (CH), E = 0 if the flow is free surface (SL) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 18 / 50

  15. The PFS model The « mixed » variable We introduce a state indicator � 1 if the flow is pressurized (CH), E = 0 if the flow is free surface (SL) and the physical section of water S by : � S if E = 1 , S = S ( A sl , E ) = A sl if E = 0 . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 18 / 50

  16. The PFS model The « mixed » variable We introduce a state indicator � 1 if the flow is pressurized (CH), E = 0 if the flow is free surface (SL) and the physical section of water S by : � S if E = 1 , S = S ( A sl , E ) = A sl if E = 0 . We set � S ( A sl , 0) = A sl if SL ρ ¯ ρ ¯ A = S = : the « mixed » variable S ( A sl , 1) = A ch if CH ρ 0 ρ 0 Q = Au : the discharge M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 18 / 50

  17. The PFS model The « mixed » variable We introduce a state indicator � 1 if the flow is pressurized (CH), E = 0 if the flow is free surface (SL) and the physical section of water S by : � S if E = 1 , S = S ( A sl , E ) = A sl if E = 0 . We set � S ( A sl , 0) = A sl if SL ρ ¯ ρ ¯ A = S = : the « mixed » variable S ( A sl , 1) = A ch if CH ρ 0 ρ 0 Q = Au : the discharge Continuity of S at transition point M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 18 / 50

  18. The PFS model Construction of the « mixed » pressure Continuity of S = ⇒ continuity of p at transition point − → p ( x, A, E ) = c 2 ( A − S ) + gI 1 ( x, S ) cos θ M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 19 / 50

  19. The PFS model Construction of the « mixed » pressure Continuity of S = ⇒ continuity of p at transition point − → p ( x, A, E ) = c 2 ( A − S ) + gI 1 ( x, S ) cos θ Similar construction for the pressure source term : � d S � A Pr ( x, A, E ) = c 2 S − 1 dx + gI 2 ( x, S ) cos θ M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 19 / 50

  20. The PFS model  ∂ t ( A ) + ∂ x ( Q ) = 0      � Q 2 �    = − g A d   ∂ t ( Q ) + ∂ x A + p ( x, A, E ) dxZ ( x )     + Pr ( x, A, E )      − G ( x, A, E )         − g K ( x, S ) Q | Q |  A C. Bourdarias, M. Ersoy and S. Gerbi A model for unsteady mixed flows in non uniform closed water pipes. SCIENCE CHINA Mathematics , 55(1) :1–26, 2012. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 20 / 50

  21. 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 � u 2 � 2 + c 2 ln( A/S ) + g H ( S ) cos θ + g Z ∂ t u + ∂ x = − g K ( x, S ) u | u | and for u = 0 , we have : c 2 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 ) = Q 2 2 A + c 2 A ln( A/ S ) + c 2 S + gz ( x, S ) cos θ + gAZ which satisfies ∂ t E + ∂ x ( E u + p ( x, A, E ) u ) = − g A K ( x, S ) u 2 | u | � 0 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 21 / 50

  22. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 22 / 50

  23. Finite Volume (VF) numerical scheme of order 1 PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 23 / 50

  24. Finite Volume (VF) numerical scheme of order 1 PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 23 / 50

  25. Finite Volume (VF) numerical scheme of order 1 PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i Cell-centered numerical scheme : i − ∆ t n � � U n +1 = U n + ∆ t n S ( U n F i +1 / 2 − F i − 1 / 2 i ) i ∆ x where � t n +1 � ∆ t n S n i ≈ S ( t, x ) dx dt t n m i M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 23 / 50

  26. Finite Volume (VF) numerical scheme of order 1 PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i Upwinded numerical scheme : � � i − ∆ t n U n +1 = U n F i +1 / 2 − � � F i − 1 / 2 i ∆ x F and � F are consistent. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 23 / 50

  27. Choice of the numerical fluxes Our goal : define F i +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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 24 / 50

  28. Choice of the numerical fluxes Our goal : define F i +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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 24 / 50

  29. Choice of the numerical fluxes Our goal : define F i +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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 24 / 50

  30. Choice of the numerical fluxes Our goal : define F i +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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 24 / 50

  31. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 25 / 50

  32. Philosophy As in kinetic theory of gases, Describe the macroscopic behavior from particle motions, here, assumed fictitious � a χ density function and by introducing a M ( t, x, ξ ; χ ) maxwellian function (or a Gibbs equilibrium) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 26 / 50

  33. Philosophy As in kinetic theory of gases, Describe the macroscopic behavior from particle motions, here, assumed fictitious � a χ density function and by introducing a M ( t, x, ξ ; χ ) maxwellian function (or a Gibbs equilibrium) i.e., transform the nonlinear system into a kinetic transport equation on M . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 26 / 50

  34. Philosophy As in kinetic theory of gases, Describe the macroscopic behavior from particle motions, here, assumed fictitious � a χ density function and by introducing a M ( t, x, ξ ; χ ) maxwellian function (or a Gibbs equilibrium) i.e., transform the nonlinear system into a kinetic transport equation on M . Thus, to be able to define the numerical macroscopic fluxes from the microscopic one. ... Faire d’une pierre deux coups ... M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 26 / 50

  35. Principle Density function We introduce � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 27 / 50

  36. Principle Gibbs Equilibrium or Maxwellian We introduce � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R then we define the Gibbs equilibrium by � ξ − u ( t, x ) � M ( t, x, ξ ) = A ( t, x ) b ( t, x ) χ b ( t, x ) with � p ( t, x ) b ( t, x ) = A ( t, x ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 27 / 50

  37. Principle Since � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R and � ξ − u ( t, x ) � M ( t, x, ξ ) = A ( t, x ) b ( t, x ) χ b ( t, x ) then micro-macroscopic relations � A = M ( t, x, ξ ) dξ � R Q = ξ M ( t, x, ξ ) dξ � R Q 2 A + A b 2 ξ 2 M ( t, x, ξ ) dξ = ���� R p M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 27 / 50

  38. Principle [P02] The kinetic formulation ( A, Q ) is solution of the PFS system if and only if M satisfy the transport equation : ∂ t M + ξ · ∂ x M − g Φ ∂ ξ M = K ( t, x, ξ ) where K ( t, x, ξ ) is a collision kernel satisfying a.e. ( t, x ) � � K dξ = 0 , ξ K d ξ = 0 R R 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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 28 / 50

  39. Principe The kinetic formulation ( A, Q ) is solution of the PFS system if and only if M satisfy the transport equation : ∂ t M + ξ · ∂ x M − g Φ ∂ ξ M = K ( t, x, ξ ) where K ( t, x, ξ ) is a collision kernel satisfying a.e. ( t, x ) � � K dξ = 0 , ξ K d ξ = 0 R R and Φ are the source terms. General form of the source terms : conservative non conservative friction � �� � ���� � �� � d B · d K Q | Q | Φ = dxZ + dx W + A 2 with W = ( Z, S, cos θ ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 28 / 50

  40. Discretization of source terms Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 29 / 50

  41. Discretization of source terms Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 29 / 50

  42. Simplification of the transport equation Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W = ⇒ ∂ t M + ξ · ∂ x M = K ( t, x, ξ ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 29 / 50

  43. Simplification of the transport equation Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W = ⇒  ∂ t f + ξ · ∂ x f = 0  � ξ − u ( t n , x, ξ ) � := A ( t n , x, ξ ) def f ( t n , x, ξ ) = M ( t n , x, ξ ) b ( t n , x, ξ ) χ  b ( t n , x, ξ ) by neglecting the collision kernel. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 29 / 50

  44. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 30 / 50

  45. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 30 / 50

  46. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x where � � � � � A n +1 1 def U n +1 f n +1 i = := ( ξ ) dξ i Q n +1 i ξ i R M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 30 / 50

  47. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x or � � i − ∆ t n � i +1 / 2 − � U n +1 = U n F − F + i i − 1 / 2 ∆ x with � 1 � � F ± � M ± 2 = ξ 2 ( ξ ) dξ. i ± 1 i ± 1 ξ R M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 30 / 50

  48. The microscopic fluxes Interpretation : potential bareer positive transmission � �� � M − 1 { ξ> 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 31 / 50

  49. The microscopic fluxes Interpretation : potential bareer positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 31 / 50

  50. The microscopic fluxes Interpretation : potential bareer positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission ∆Φ n i +1 / 2 may be interpreted as a time-dependent slope ! M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 31 / 50

  51. The microscopic fluxes Interpretation : Dynamic slope = ⇒ Upwinding of the friction positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission ∆Φ n i +1 / 2 may be interpreted as a time-dependent slope ! . . . we reintegrate the friction . . . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 31 / 50

  52. Upwinding of the source terms : ∆Φ i +1 / 2 conservative ∂ x W : W i +1 − W i non-conservative B ∂ x W : B ( W i +1 − W i ) where � 1 B = B ( s, φ ( s, W i , W i +1 )) ds 0 for the « straight lines paths » , i.e. φ ( s, W i , W i +1 ) = s W i +1 + (1 − s ) W i , 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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 32 / 50

  53. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 33 / 50

  54. χ =??? in practice ? ? ? Let us recall that we have to define a χ function such that : � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R � ξ − u � and M = A b χ satisfies the equation : b ∂ t M + ξ · ∂ x M − g Φ ∂ ξ M = 0 and χ − → definition of the macroscopic fluxes. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 34 / 50

  55. Properties related to χ We always have Conservativity of A holds for every χ . Positivity of A holds for every χ but for numerical purpose iff supp χ is compact to get a CFL condition. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 35 / 50

  56. Properties related to χ We always have Conservativity of A holds for every χ . Positivity of A holds for every χ but for numerical purpose iff supp χ is compact to get a CFL condition. while discrete equilibrium, discrete entropy inequalities strongly depend on the choice of the χ function. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 35 / 50

  57. Properties related to χ We always have Conservativity of A holds for every χ . Positivity of A holds for every χ but for numerical purpose iff supp χ is compact to get a CFL condition. while discrete equilibrium, discrete entropy inequalities strongly depend on the choice of the χ function. In the following, we only focus on discrete equilibrium. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 35 / 50

  58. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 36 / 50

  59. Strategy Even if the pipe is circular with uniform cross-sections, for instance for the free surface flows, the following procedure fails for complex source terms : Following [PS01], choose χ such that M ( t, x, ξ ; χ ) is the steady state solution at rest, u = 0 : ξ · ∂ x M − g Φ ∂ ξ M = 0 . provides � A 2 � − w 2 A 2 − I 1 T 3 T I 1 − A 2 χ ′ ( w ) = 0 where w = ξ wχ ( w ) + b . 2 I 1 I 1 2 I 1 B. Perthame and C. Simeoni A kinetic scheme for the Saint-Venant system with a source term. Calcolo , 38(4) :201–231, 2001. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 37 / 50

  60. Strategy Even if the pipe is circular with uniform cross-sections, for instance for the free surface flows, the following procedure fails for complex source terms : Following [PS01], choose χ such that M ( t, x, ξ ; χ ) is the steady state solution at rest, u = 0 : ξ · ∂ x M − g Φ ∂ ξ M = 0 . provides           − w 2 A 2 − I 1 T 3 T I 1 − A 2 A 2 χ ′ ( w ) = 0 . wχ ( w ) + 2 I 1  I 1 2 I 1    � �� �  ���� � �� �    α γ β Then, this equation is solvable as an ODE iff the coefficients ( α, β, γ ) are constants. B. Perthame and C. Simeoni A kinetic scheme for the Saint-Venant system with a source term. Calcolo , 38(4) :201–231, 2001. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 37 / 50

  61. Strategy Even if the pipe is circular with uniform cross-sections, for instance for the free surface flows, the following procedure fails for complex source terms : Following [PS01], choose χ such that M ( t, x, ξ ; χ ) is the steady state solution at rest, u = 0 : ξ · ∂ x M − g Φ ∂ ξ M = 0 . provides           − w 2 A 2 − I 1 T 3 T I 1 − A 2 A 2 χ ′ ( w ) = 0 . wχ ( w ) + 2 I 1  I 1 2 I 1    � �� �  ���� � �� �    α γ β Then, this equation is solvable as an ODE iff the coefficients ( α, β, γ ) are constants. � T � 2 , 2 T, T For a rectangular pipe with uniform sections, we have ( α, β, γ ) = 2 with T = cst the base of the pipe. B. Perthame and C. Simeoni A kinetic scheme for the Saint-Venant system with a source term. Calcolo , 38(4) :201–231, 2001. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 37 / 50

  62. In these settings � T � 2 , 2 T, T With ( α, β, γ ) = and 2 Theorem � � 1 / 2 1 − w 2 we get χ ( w ) = 1 and the numerical scheme satisfies the following π 4 + properties : Positivity of A (under a CFL condition), Conservativity of A , Discrete equilibrium, Discrete entropy inequalities. This results holds only for conservative terms ∂ x Z ( x ) . A similar result for pressurized flows, unusable in practice (see [PhDErsoy] Chap. 2). M. Ersoy Modeling, mathematical and numerical analysis of various compressible or incompressible flows in thin layer [Mod´ elisation, analyse math´ ematique et num´ erique de divers ´ ecoulements compressibles ou incompressibles en couche mince]. Universit´ e de Savoie, Chamb´ ery , September 10, 2010. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 38 / 50

  63. If ( α, β, γ ) are not constants . . . Then, the equation to solve is : ξ · ∂ x M − g Φ ∂ ξ M = 0 . Complicate to solve − → find an easy way to maintain, at least, discrete steady states. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 39 / 50

  64. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 40 / 50

  65. Correction of the macroscopic fluxes The steady state is perfectly maintained iff F − � i +1 / 2 ( U i , U i +1 , Z i , Z i +1 ) − � F + i − 1 / 2 ( U i − 1 , U i , Z i − 1 , Z i ) = 0 with U = ( A, Q ) , Z = ”source terms” � Notations : Fi ± 1 / 2 the numerical flux of the homogeneous system, Fi ± 1 / 2 the numerical flux with source term and F the flux of the PFS-model. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 41 / 50

  66. Correction of the macroscopic fluxes The steady state is perfectly maintained iff F − � i +1 / 2 ( U i , U i +1 , Z i , Z i +1 ) − � F + i − 1 / 2 ( U i − 1 , U i , Z i − 1 , Z i ) = 0 with U = ( A, Q ) , Z = ”source terms” Let us recall that without sources, whenever the numerical flux is consistent, i.e. ∀ U = ( A, Q ) ∈ R 2 , F i ± 1 / 2 ( U , U ) = F ( U ) , we automatically have, whenever steady states occurs : F − i +1 / 2 ( U i , U i +1 ) − F + i − 1 / 2 ( U i − 1 , U i ) = 0 , i.e., U n +1 = U n i . i � Notations : Fi ± 1 / 2 the numerical flux of the homogeneous system, Fi ± 1 / 2 the numerical flux with source term and F the flux of the PFS-model. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 41 / 50

  67. Correction of the macroscopic fluxes The steady state is perfectly maintained iff F − � i +1 / 2 ( U i , U i +1 , Z i , Z i +1 ) − � F + i − 1 / 2 ( U i − 1 , U i , Z i − 1 , Z i ) = 0 with U = ( A, Q ) , Z = ”source terms” Let us recall that without sources, whenever the numerical flux is consistent, i.e. ∀ U = ( A, Q ) ∈ R 2 , F i ± 1 / 2 ( U , U ) = F ( U ) , we automatically have, whenever steady states occurs : F − i +1 / 2 ( U i , U i +1 ) − F + i − 1 / 2 ( U i − 1 , U i ) = 0 , i.e., U n +1 = U n i . i Correction of the numerical flux → toward a well balanced scheme � Notations : Fi ± 1 / 2 the numerical flux of the homogeneous system, Fi ± 1 / 2 the numerical flux with source term and F the flux of the PFS-model. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 41 / 50

  68. Definition of the new fluxes : M-scheme IDEAS : replace dynamic quantities U i − 1 and U i +1 by stationary profiles U + i − 1 and U − i +1 sources terms Z i − 1 and Z i +1 by stationary profiles Z + i − 1 and Z − i +1 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 42 / 50

  69. Definition of the new fluxes : M-scheme IDEAS : replace dynamic quantities U i − 1 and U i +1 by stationary profiles U + i − 1 and U − i +1 sources terms Z i − 1 and Z i +1 by stationary profiles Z + i − 1 and Z − i +1 With A − i +1 and A + i − 1 computed from the steady states : � D ( A − i +1 , Q i +1 , Z i ) = D ( U i +1 , Z i +1 ) ∀ i, D ( A + i − 1 , Q i − 1 , Z i ) = D ( U i − 1 , Z i − 1 )  g H ( A ) cos θ + gZ if E = 0 ,  where D ( U , Z ) = Q 2 � A � 2 A + c 2 ln + g H ( S ) cos θ + gZ if E = 1 .  S And ( Z − i +1 , Z + i − 1 ) are defined as follows : � Z i A − if i +1 = A i Z − i +1 = A − Z i +1 if i +1 � = A i � Z i A + if i − 1 = A i Z + i − 1 = A + if i − 1 � = A i Z i − 1 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 42 / 50

  70. Definition of the new fluxes : M-scheme IDEAS : replace dynamic quantities U i − 1 and U i +1 by stationary profiles U + i − 1 and U − i +1 sources terms Z i − 1 and Z i +1 by stationary profiles Z + i − 1 and Z − i +1 Let us now consider U n +1 = U n i + i � � ∆ t n F − 2 ( U n i , A − i +1 , Z i , Z − i +1 ) − F + 2 ( A + i − 1 , U n i , Z + i +1 , Q n i − 1 , Q n i − 1 , Z i ) i + 1 i − 1 ∆ x M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 42 / 50

  71. Definition of the new fluxes : M-scheme IDEAS : replace dynamic quantities U i − 1 and U i +1 by stationary profiles U + i − 1 and U − i +1 sources terms Z i − 1 and Z i +1 by stationary profiles Z + i − 1 and Z − i +1 Let us now consider U n +1 = U n i + i � � ∆ t n F − 2 ( U n i , A − i +1 , Q n i +1 , Z i , Z − i +1 ) − F + 2 ( A + i − 1 , Q n i − 1 , U n i , Z + i − 1 , Z i ) i + 1 i − 1 ∆ x instead of the previous one : U n +1 = U n i + i � � ∆ t n F − 2 ( U n i +1 , Z i , Z i +1 ) − F + i − 1 , U n i , A n i +1 , Q n 2 ( A n i − 1 , Q n i , Z i − 1 , Z i ) i + 1 i − 1 ∆ x M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 42 / 50

  72. Definition of the new fluxes : M-scheme IDEAS : replace dynamic quantities U i − 1 and U i +1 by stationary profiles U + i − 1 and U − i +1 sources terms Z i − 1 and Z i +1 by stationary profiles Z + i − 1 and Z − i +1 Let us now consider U n +1 = U n i + i � � ∆ t n F − 2 ( U n i , A − i +1 , Z i , Z − i +1 ) − F + 2 ( A + i − 1 , U n i , Z + i +1 , Q n i − 1 , Q n i − 1 , Z i ) i + 1 i − 1 ∆ x Then, Theorem the numerical scheme is well-balanced. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 42 / 50

  73. Proof the numerical flux is, by construction, consistent. M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 43 / 50

  74. Proof the numerical flux is, by construction, consistent. Let us assume that there exits n such that for every i : i = Q 0 , D ( U n Q n i , Z i ) = h 0 . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 43 / 50

  75. Proof the numerical flux is, by construction, consistent. Let us assume that there exits n such that for every i : i = Q 0 , D ( U n Q n i , Z i ) = h 0 . Then, D ( A − i +1 , Q i +1 , Z i ) = D ( U i +1 , Z i +1 ) = h 0 , ∀ i and especially, we have : D ( A − i +1 , Q i +1 , Z i ) = D ( U i , Z i ) . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 43 / 50

  76. Proof the numerical flux is, by construction, consistent. Let us assume that there exits n such that for every i : i = Q 0 , D ( U n Q n i , Z i ) = h 0 . Then, D ( A − i +1 , Q i +1 , Z i ) = D ( U i +1 , Z i +1 ) = h 0 , ∀ i and especially, we have : D ( A − i +1 , Q i +1 , Z i ) = D ( U i , Z i ) . The application A → D ( A, Q, Z ) being injective, provides A − i +1 = A i and thus Z − i +1 = Z i by construction. Similarly, we get A + i − 1 = A i and Z + i − 1 = Z i . M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 43 / 50

  77. Proof the numerical flux is, by construction, consistent. Let us assume that there exits n such that for every i : i = Q 0 , D ( U n Q n i , Z i ) = h 0 . Then, D ( A − i +1 , Q i +1 , Z i ) = D ( U i +1 , Z i +1 ) = h 0 , ∀ i and especially, we have : D ( A − i +1 , Q i +1 , Z i ) = D ( U i , Z i ) . The application A → D ( A, Q, Z ) being injective, provides A − i +1 = A i and thus Z − i +1 = Z i by construction. Similarly, we get A + i − 1 = A i and Z + i − 1 = Z i . Finally, since F − 2 ( U n i , U − i +1 , Z i , Z − i +1 ) − F + 2 ( U + i − 1 , U n i , Z + i − 1 , Z i ) = 0 , i + 1 i − 1 we get ∀ l � n, Q l +1 = Q l i := Q 0 . i � M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 43 / 50

  78. Numerical properties For instance, with the simplest χ function [ABP00], 1 χ ( ω ) = √ √ √ 3] ( ω ) 3 1 [ − 3 , 2 the following properties holds : Positivity of A (under a CFL condition), Conservativity of A , Discrete equilibrium and, Natural treatment of drying and flooding area. for example and analytical expression of the numerical macroscopic 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 (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 44 / 50

  79. Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Previous works Formal derivation of the free surface and pressurized model A coupling : the PFS-model 2 A Finite Volume Framework Kinetic Formulation and numerical scheme The χ function and well balanced scheme 1. Classical scheme fails in presence of complex source terms 2. An alternative toward a Well-Balanced scheme Numerical results 3 Conclusion and perspectives M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 45 / 50

  80. Qualitative analysis of convergence and comparison with the Well-Balanced VFRoe scheme T = 0.000 Eau Ligne piezometrique 104 103 m d’eau 102 101 100 99 0 100 200 300 400 500 600 700 800 900 m upstream piezometric head 104 m Niveau piezometrique aval 103.2 103 102.8 102.6 m d’eau 102.4 102.2 102 101.8 101.6 Hauteur piezo haut du tuyau 101.4 0 2 4 6 8 10 12 14 downstream piezometric head : Temps (s) M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 46 / 50

  81. Convergence During unsteady flows t = 100 s Erreur L2 : Ligne piezometrique au temps t = 100 s 0.8 Ordre VFRoe (polyfit) = 0.91301 VFRoe (sans polyfit) Ordre FKA (polyfit) = 0.88039 0.6 FKA (sans polyfit) 0.4 0.2 0 � y � L 2 -0.2 -0.4 -0.6 -0.8 -1 ln(∆ x ) 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 M. Ersoy (IMATH) A Well Balanced Finite Volume Kinetic scheme IMATH 46 / 50

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