a well balanced finite volume kinetic fvk scheme for
play

A Well Balanced Finite Volume Kinetic (FVK) scheme for unsteady - PowerPoint PPT Presentation

May 29, 2023 •179 likes •1.26k views

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

Recommend

On a well-balanced high-order finite volume scheme for the shallow water equations with bottom

On a well-balanced high-order finite volume scheme for the shallow water equations with bottom

On a well-balanced high-order finite volume scheme for the shallow water equations with bottom topography and dry areas Jos M. Gallardo, Manuel J. Castro, Carlos Pars and Jos M. Gonzlez-Vida Department of Mathematical Analysis,

501 views • 20 slides
Unsteady mixed flows in closed water pipes. A well-balanced finite volume scheme Christian

Unsteady mixed flows in closed water pipes. A well-balanced finite volume scheme Christian

Unsteady mixed flows in closed water pipes. A well-balanced finite volume scheme Christian Bourdarias, Mehmet Ersoy and Stphane Gerbi LAMA, Universit de Savoie, Chambry, France 2 nd Workshop Mathematics and Oceanography Montpellier, 1-2-3

757 views • 75 slides
HYP 2012 A positive, well-balanced and entropy-satisfying scheme for shallow water flows

HYP 2012 A positive, well-balanced and entropy-satisfying scheme for shallow water flows

HYP 2012 A positive, well-balanced and entropy-satisfying scheme for shallow water flows Interest of the kinetic description E. Audusse, M.-O. Bristeau, C. Pares &amp; J. Sainte-Marie Padova - june 2012 Introduction Kinetic description &amp;

327 views • 20 slides
A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite

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

A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme M. Ersoy 1, Christian Bourdarias 2 and St ephane Gerbi 3 LMB, Besan con, the 10 February 2011 1. BCAM, Spain, mersoy@bcamath.org 2.

825 views • 68 slides
High-order well-balanced finite-volume schemes for eddy computations in barostrophic jets.

High-order well-balanced finite-volume schemes for eddy computations in barostrophic jets.

High-order well-balanced finite-volume schemes for eddy computations in barostrophic jets. Algorithms and numerical comparisons Normann Pankratz IGPM RWTH Aachen Normann Pankratz (Aachen) High-order well-balanced Finite-Volume 1 / 38

732 views • 54 slides
A Finite Volume Kinetic (FVK) framework for unsteady mixed flows in non uniform closed water

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

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,

796 views • 51 slides
A well-balanced scheme for a 2D finite difference, non-hydrostatic atmospheric model Andreas

A well-balanced scheme for a 2D finite difference, non-hydrostatic atmospheric model Andreas

A well-balanced scheme for a 2D finite difference, non-hydrostatic atmospheric model Andreas Dobler 22/09/06 Thanks to: Christoph Schr, Jrg Schmidli, Stefan Wunderlich 22/09/06 Andreas Dobler / Institute for Atmospheric and Climate

240 views • 23 slides
Finite volume method for linear and non linear elliptic problems with discontinuities Franck

Finite volume method for linear and non linear elliptic problems with discontinuities Franck

Finite volume method for linear and non linear elliptic problems with discontinuities Franck BOYER and Florence HUBERT L.A.T.P. - Marseille, FRANCE Paris December 2007 1/ 46 O UTLINES I NTRODUCTION The classical finite volume scheme

948 views • 60 slides
A Finite Volume Evolution Galerkin Scheme for Wave Propagation in Heterogeneous Media K. R. Arun

A Finite Volume Evolution Galerkin Scheme for Wave Propagation in Heterogeneous Media K. R. Arun

A Finite Volume Evolution Galerkin Scheme for Wave Propagation in Heterogeneous Media K. R. Arun Institut f ur Geometrie und Praktische Mathematik, RWTH Aachen, Germany 14th International Conference on Hyperbolic Problems: Theory, Numerics

439 views • 30 slides
Convergence of a Finite Volume Scheme and Numerical Simulations for Water-Gas Flow in Porous

Convergence of a Finite Volume Scheme and Numerical Simulations for Water-Gas Flow in Porous

Convergence of a Finite Volume Scheme and Numerical Simulations for Water-Gas Flow in Porous Media M. Afif and B. Amaziane IPRALMA de Pau CNRS-UMR 5142 brahim.amaziane@univ-pau.fr Journ ee MoMaS : Mod elisation des Ecoulements

379 views • 18 slides
Very high-order finite volume scheme for the 2D linear convection-diffusion problem S. Clain 1 , 3

Very high-order finite volume scheme for the 2D linear convection-diffusion problem S. Clain 1 , 3

Very high-order finite volume scheme for the 2D linear convection-diffusion problem S. Clain 1 , 3 , G.J. Machado 1 , J.M. Nobrega 2 , R.M.S. Pereira 1 , A. Boularas 4 1 Mathematical Centre, University of Minho, Portugal 2 Institute for Polymers

378 views • 33 slides
A variational finite volume scheme for Wasserstein gradient flows es 1 , T. O. Gallou et 2 , G.

A variational finite volume scheme for Wasserstein gradient flows es 1 , T. O. Gallou et 2 , G.

A variational finite volume scheme for Wasserstein gradient flows es 1 , T. O. Gallou et 2 , G. Todeschi 2 C. Canc` Inria Lille, Rapsodi 1 Inria Paris, MOKAPLAN 2 ICODE, January 8, 2010 Wasserstein gradient flows domain R d convex,

609 views • 30 slides
A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum

A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum

1 A second order finite volume scheme on space-adaptive staggered grids in 3D Wolfram Rosenbaum Prof. Dr. Sebastian Noelle RWTH Aachen, Germany Staggered grid construction in 1D 2 How to construct dual cells dual cell dual grid primal

510 views • 21 slides
A Well-Balanced Multi-Dimensional Reconstruction Scheme for Hydrostatic Equilibria Roger

A Well-Balanced Multi-Dimensional Reconstruction Scheme for Hydrostatic Equilibria Roger

A Well-Balanced Multi-Dimensional Reconstruction Scheme for Hydrostatic Equilibria Roger Kppeli Outline Introduction (Astro)Physical motivation Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second

418 views • 37 slides
Finite volume method for linear and non linear elliptic problems with discontinuities Franck

Finite volume method for linear and non linear elliptic problems with discontinuities Franck

Finite volume method for linear and non linear elliptic problems with discontinuities Franck BOYER et Florence HUBERT L.A.T.P . - Marseille, FRANCE DD17 - 2006 1/ 37 Plan P LAN 1 I NTRODUCTION 2 T HE DDFV SCHEME 3 T HE 1D PROBLEM 4 T HE 2D

400 views • 39 slides
A finite volume method on general meshes for a time evolution convection-diffusion equation.

A finite volume method on general meshes for a time evolution convection-diffusion equation.

A finite volume method on general meshes for a time evolution convection-diffusion equation. Konstantin Brenner Supervised by Danielle Hilhorst University Paris-Sud XI MoMaS Model problem &amp; Numerical scheme ; Existence of the unique

603 views • 29 slides
A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind

A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind

Modelisation Kinetic approach Tests Conclusion A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind all the source terms C. B OURDARIAS M. E RSOY S. G ERBI LAMA-Universit de Savoie de Chambry,

811 views • 31 slides
A conservative well-balanced hybrid SPH scheme for the shallow-water model C. Berthon 1 , M. de

A conservative well-balanced hybrid SPH scheme for the shallow-water model C. Berthon 1 , M. de

A conservative well-balanced hybrid SPH scheme for the shallow-water model A conservative well-balanced hybrid SPH scheme for the shallow-water model C. Berthon 1 , M. de Leffe 2 , V. Michel-Dansac 1 1 Laboratoire de Math ematiques Jean Leray,

602 views • 58 slides
CEMRACS 2019 Coupling model of underground flow and pollution transport using a Finite volume

CEMRACS 2019 Coupling model of underground flow and pollution transport using a Finite volume

CEMRACS 2019 Coupling model of underground flow and pollution transport using a Finite volume scheme Ayoub Charhabil charhabil@math.univ-paris13.fr F. Benkhaldoun Paris 13 University August 8th, 2019 A.Charhabil Paris 13 IRSN 1 / 45

718 views • 45 slides
A kinetic scheme for air entrainment in transient flows: a two-layer approach. Mehmet Ersoy 1

A kinetic scheme for air entrainment in transient flows: a two-layer approach. Mehmet Ersoy 1

A kinetic scheme for air entrainment in transient flows: a two-layer approach. Mehmet Ersoy 1 joint work with C. Bourdarias and S. Gerbi, LAMA, UMR 5127 CNRS, Universit e de Savoie Mont-Blanc, Chamb ery, France Dynamics and modeling of

1.07k views • 83 slides
Towards an ultra efficient kinetic scheme : High-Performance Computing G. Dimarco 1 J. Narski 2

Towards an ultra efficient kinetic scheme : High-Performance Computing G. Dimarco 1 J. Narski 2

Towards an ultra efficient kinetic scheme : High-Performance Computing G. Dimarco 1 J. Narski 2 ere 2 R. Loub` 1 Department of Mathematics and Computer Science, University of Ferrara, Italy. 2 Institut de Math ematiques de Toulouse, Universit

678 views • 40 slides
Balanced- -energy Sleep energy Sleep Balanced Scheduling Scheme for High Scheduling Scheme for

Balanced- -energy Sleep energy Sleep Balanced Scheduling Scheme for High Scheduling Scheme for

Balanced- -energy Sleep energy Sleep Balanced Scheduling Scheme for High Scheduling Scheme for High Density Cluster- -based Sensor based Sensor Density Cluster Networks Networks and P. J. Deng * , Y. Han ** , W. Heinzelman and P.

226 views • 18 slides
Finite volume methods for dissipative problems Claire Chainais-Hillairet CEMRACS summer school

Finite volume methods for dissipative problems Claire Chainais-Hillairet CEMRACS summer school

Finite volume methods for dissipative problems Claire Chainais-Hillairet CEMRACS summer school Marseille, July 2019, 15-20 Lecture 3 : Finite volume schemes and long time behavior Outline Discrete functional inequalities 1 Results for the

586 views • 54 slides
An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations

An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations

An asymptotic preserving unified gas kinetic scheme for the grey radiative transfer equations Song Jiang Institute of Applied Physics and Computational Mathematics, Beijing China-Gemany Workshop, May 23-26, 2014, Beijing (joint work with

534 views • 31 slides

More recommend