Skew-Symmetric Schemes for compressible and incompressible flows

Size: px

Start display at page:

Download "Skew-Symmetric Schemes for compressible and incompressible flows"

Shauna Page
6 years ago
Views:

1 Skew-Symmetric Schemes for compressible and incompressible flows Julius Reiss Fachgebiet für Numerische Fluiddynamik TU Berlin CEMRACS J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, 202 / 46

2 shock acoustic Use Finite Difference or use Finite Volume? J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8th, / 46

3 shock acoustic Use Finite Difference or use Finite Volume? skew symmetric, conservative FD J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8th, / 46

4 Overview Burgers equation 2 Euler/Navier-Stokes Equations 3 Time discretization 4 Arbitrarily Transformed Grids 5 Fluxes & Boundary conditions 6 Incompressible flows J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

5 The concept t u + x f (u) = 0 Discretised: t u + D u u = 0 t u dx T D u = 0 telescoping sum t u 2 dx (D u ) T = D u skew symmetry J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

6 The concept t u + x f (u) = 0 Discretised: t u + D u u = 0 t u dx T D u = 0 telescoping sum Momentum Conservation t u 2 dx (D u ) T = D u skew symmetry Kin. Energy Conservation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

7 Literature: Skew Symmetry Feiereisen W.C.Reynolds J.H. Ferziger, Numerical simulation of a compressible homogeneous, turbulent shear flow, NASA-CR-64953; SU-TF-3 E. Tadmor, Skew-Selfadjoint Form for Systems of Conservation Laws, J. Math. Ana. Appl. 03, p428, (984) Y. Morinishi and T. S. Lund and O. V. Vasilyev and P. Moin, Fully conservative higher order finite difference schemes for incompressible flow JCP 43, p 90, (998) R. W. C. P. Verstappen, A. E. P. Veldman. Symmetry-preserving discretization of turbulent flow. JCP 87, p343, (2003) J.C. Kok, A high-order low-dispersion symmetry-preserving finite-volume method... JCP 228, p 68, (2009) Y. Morinishi, Skew-symmetric form of convective terms..., JCP 229, p276 (200) S. Pirozzoli, Generalized conservative approximations of split convective derivative operators, p780 JCP 229, (200) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

8 Skew Symmetric discretisation Simple example: Burgers equation divergence and convection form by (2 [D] + [C])/3: skew symmetric form ( ) t u+ u 2 x 2 = 0 (D) t u+ u x u = 0 (C) t u + 3 [ xu +u x ]u = 0 (S) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

9 Skew Symmetric discretisation Burgers in skew symmetric form t u + 3 [ xu +u x ]u = 0 Discrete t u + (DU + UD) u = 0 } 3 {{} D u U = diag(u), derivative D with D T = D t u + D u u = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

10 Skew Symmetric discretisation Kinetic Energy Conservation 2 ut u = 2 i u2 i t u T u = ( t u) T u + u T t u = (D u u) T u u T D u u = u T [ (D u ) T + D u] u J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

11 Skew Symmetric discretisation Kinetic Energy Conservation 2 ut u = 2 i u2 i t u T u = ( t u) T u + u T t u = (D u u) T u u T D u u = u T [ (D u ) T + D u] u Symmetry of transport term D u (D u ) T = 3 (DU + UD)T = 3 (UT D T + D T U T ) = D u U = diag(u) = U T, D T = D J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

12 Skew Symmetric discretisation Kinetic Energy Conservation 2 ut u = 2 i u2 i Symmetry of transport term D u t u T u = ( t u) T u + u T t u = (D u u) T u u T D u u = u T [ (D u ) T + D u ] u = 0 }{{} =0 (D u ) T = 3 (DU + UD)T = 3 (UT D T + D T U T ) = D u Skew symmetry implies conservation of E kin J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

13 Skew Symmetric discretisation Momentum Conservation T u = i u i Telescoping with D T = D t T u = T t u = T D u u T D u u = 3 (T }{{} D U + T UD)u = 3 } ut {{ Du } = 0 =0 =0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

14 Skew Symmetric discretisation Momentum Conservation T u = i u i Telescoping t T u = T t u = } T {{ D u u } = 0 =0 T D u u = 3 (T }{{} D U + T UD)u = 3 } ut {{ Du } = 0 =0 =0 Telescoping sum property implies conservation of momentum J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

15 Time Integration Implicit midpoint rule Fully discrete u n+ u n t with u n+/2 = 2 (un + u n+ ) + 3 Dun+/2 u n+/2 = 0 Verstappen, Veldman, J. Com. Phys 87, p. 343 (2003) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

16 Time Integration Implicit midpoint rule Fully discrete u n+ u n t with u n+/2 = 2 (un + u n+ ) + 3 Dun+/2 u n+/2 = 0 Multiplying by (u n+/2 ) T (u n+/2 ) T (u n+ u n ) + 3 (un+/2 ) T D un+/2 u n+/2 }{{} =0 = 2 (un + u n+ ) T (u n+ u n ) = (un+ ) 2 2 (un ) 2 2 = 0 Verstappen, Veldman, J. Com. Phys 87, p. 343 (2003) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

17 Numerical example: Burgers Equation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, 202 / 46

18 Overview Burgers equation 2 Euler/Navier-Stokes Equations 3 Time discretization 4 Arbitrarily Transformed Grids 5 Fluxes & Boundary conditions 6 Incompressible flows J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

19 Skew Symmetric discretisation Euler Momentum Equation divergence and convection form t (ϱu) + x (ϱu 2 ) + x p = 0 (D) ϱ t (u) + ϱu x (u) + x p = 0 (C) by ([D]+[C])/2 Skew symmetric form 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 (S) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

20 Skew Symmetric discretisation Euler Equations t ϱ + x (uϱ) = 0 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

21 Skew Symmetric discretisation Euler Equations t ϱ + x (uϱ) = 0 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 ( ) ( ( )) p p t + x u γ γ + p + ( ) ( ( t ϱu 2 /2 + x u ϱu /2) ) 2 = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

22 Skew Symmetric discretisation Skew symmetric Euler Equations t ϱ + x (uϱ) = 0 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 γ tp + γ γ x (up) u x p = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

23 Skew Symmetric discretisation Skew symmetric Euler Equations t ϱ + x (uϱ) = 0 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 γ tp + γ γ x (up) u x p = 0 Discretised t ϱ + (DU)ϱ = 0 2 ( tϱ + ϱ t )u + (DUR + RUD)u + Dp 2 = 0 γ tp + γ (DU)p (UD)p γ = 0 with U = diag(u), R = diag(ϱ), Derivative D = D T J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

24 Skew Symmetric discretisation Skew symmetric Navier Stokes Equations t ϱ + x (uϱ) = 0 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = x τ γ tp + γ γ x (up) u x p = u x τ + x τ Discretised t ϱ + (DU)ϱ = 0 2 ( tϱ + ϱ t )u + (DUR + RUD)u + Dp 2 = Dτ γ tp + γ (DU)p (UD)p γ = UDτ + Dτu with U = diag(u), R = diag(ϱ), Derivative D = D T, τ = µ x u J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

25 Conservation, time continuous t ϱ + DUϱ = 0 2 ( tϱ + ϱ t )u + (DUR + RUD) u + Dp 2 }{{} = 0 D uϱ γ tp + γ DUp UDp γ = 0 T (mass) = 0 mass Conservation T (mom) + u T (mass)/2 = 0 Momentum Conservation T (innere) + u T (mom) = 0 Energy Conservation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

26 Conservation, time continuous t ϱ + DUϱ = 0 2 ( tϱ + ϱ t )u + (DUR + RUD) u + Dp 2 }{{} = 0 D uϱ γ tp + γ DUp UDp γ = 0 T (mass) = 0 mass Conservation T (mom) + u T (mass)/2 = 0 Momentum Conservation T (innere) + u T (mom) = 0 Energy Conservation γ t T p + 2 ut ( t ϱ + ϱ t )u = ( ) t γ T p + u T (ϱu)/2 + γ γ T DUp + 2 ut D uϱ u = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

27 Numerical example D Time Integration: Leapfrog like J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

28 Time discretization One Step methods: 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 Morinishi s rewriting 2 ( tρ +ρ t ) u = ρ t ρu use adopted midpoint rule similar to Morinishi, Subbareddy et. al., JCP 2009 generalises to higher order J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

29 Time discretization One Step methods: 2 ( tϱ +ϱ t ) u + 2 ( xuϱ +ϱu x ) u + x p = 0 Morinishi s rewriting 2 ( tρ +ρ t ) u = ρ t ρu use adopted midpoint rule similar to Morinishi, Subbareddy et. al., JCP 2009 generalises to higher order ρ t ( ρu) + 2 Duρ u + D x p = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

30 Time discretization ρ t ρ + 2 Bu ρ = 0 ρ t ( ρu) + 2 Duρ u + D x p = 0 γ tp + γ γ Bu p C u p = 0 ( ρ n+ ρ n) ρ n+/2 + t 2 Bun+a ρ n+b = 0 ( ) n+ ( ) n ρ n+/2 ρu ρu + ρ n+b t 2 Dun+a u n+/2 + D x p n+c = 0 ρ n+/2 = 2 ( ρ n + ρ n+ ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

31 Time discretization ρ t ρ + 2 Bu ρ = 0 ρ t ( ρu) + 2 Duρ u + D x p = 0 γ tp + γ γ Bu p C u p = 0 ( ρ n+ ρ n) ρ n+/2 + t 2 Bun+a ρ n+b = 0 ( ) n+ ( ) n ρ n+/2 ρu ρu + ρ n+b t 2 Dun+a u n+/2 + D x p n+c = 0 ρ n+/2 = 2 ( ρ n + ρ n+ ) u n+/2 = ( ρu) n+ +( ρu) n 2 ρ n+/2 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

32 Time discretization Higher Order ρ t ρ + 2 Bu ρ = 0 ρ t ( ρu) + 2 Duρ u + D x p = 0 γ tp + γ γ Bu p C u p = 0 Implicit Midpoint rule is Gauss-collocation method All Gauss-collocation methods preserve skew-symme. & cons. Midpoint (2 nd order) 2 2 Brouwer, Reiss, in prep. Midpoint (4 th order) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

33 Overview Burgers equation 2 Euler/Navier-Stokes Equations 3 Time discretization 4 Arbitrarily Transformed Grids 5 Fluxes & Boundary conditions 6 Incompressible flows J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

34 Euler Equations, more dimensions Skew symmetric momentum equation ρ t ( ρu α ) + 2 [ xβ ρu β +ρu β xβ ] u α + xα p = 0. Main problem: Keep skew symmetric structure on distorted grids. a Local base e α = ξ αr, r = r(ξ, η, ζ) = (x, y, z) T Use divergence as u β x β = u = J ξ α(e β e γ )u α,cy = (e β e γ ) ξ αu. J α,cy a Veldman, Rinzema, Playing with nonuniform grids, J. Engin. and Math.26, p 9, (992), VV2003 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

35 Euler Equations, more dimensions Skew symmetric momentum equation ρ t ( ρu α ) + 2 [ xβ ρu β +ρu β xβ ] u α + xα p = 0. Main problem: Keep skew symmetric structure on distorted grids. Local base e α = ξ αr, r = r(ξ, η, ζ) = (x, y, z) T Use divergence as u β x β = J ξ α(e β e γ )u α,cy = (e β e γ ) ξ αu. J α,cy y η e η e ξ ξ [ xβ ρu β + ρu β xβ ]u α x α,cy[ ξ α(e β e γ )uϱ + uϱ(e β e γ ) ξ α ]u α J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

36 Euler Equations, more dimensions Skew symmetric momentum equation ρ t ( ρu α ) + 2 [ xβ ρu β +ρu β xβ ] u α + xα p = 0. Main problem: Keep skew symmetric structure on distorted grids. Local base e α = ξ αr, r = r(ξ, η, ζ) = (x, y, z) T Use divergence as u β x β = J ξ α(e β e γ )u α,cy = (e β e γ ) ξ αu. J α,cy [ xβ ρu β + ρu β xβ ]u α y η x e η e ξ ξ α,cy[ ξ α(e β e γ )uϱ + uϱ(e β e γ ) ξ α ]u α J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

37 Euler Equations in 2D Semidiscrete J t ρ + B u ρ = 0 J ρ t ( ρu) + 2 Duρ u + D x p = 0 with J ρ t ( ρv) + 2 Duρ v + D y p = 0 J γ tp + γ γ Bu p C u p = 0 B u = D ξ Ũ + D η Ṽ D uρ = (D ξ ŨR + RŨD ξ) + (D η Ṽ R + RṼ D η) C u = UD x + VD y where Ũ = (UY η VX η ) and Ṽ = (VX ξ UY ξ ) and D x = D ξ Y η D η Y ξ, D y =... J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

38 Overview Burgers equation 2 Euler/Navier-Stokes Equations 3 Time discretization 4 Arbitrarily Transformed Grids 5 Fluxes & Boundary conditions 6 Incompressible flows J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

39 Boundary conditions Flux vs. Value, D mass equation b T = T t ρ + D x uρ = 0 t T ρ + T D x }{{} b T uρ = mass flux over boundaries: ( b T 3 uρ = (ρu) 2 (ρu) 2 Flux no-zero even for u = 0 2 ) ( 3 + (ρu) N = ( 3 2, 2, 0,, 0, 2, 3 2 ) ) 2 (ρu) N = f 0 + f N 2 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

40 Boundary conditions Would like u = 0 f 0 = 0 Wu = u. b T = (, 0,..., 0, ). weigh- matrix W (norm) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

41 Boundary conditions Would like u = 0 f 0 = 0 Wu = u. b T = (, 0,..., 0, ). weigh- matrix W (norm) W ij = diag( 2,,,...,, 2 ) Summation By Parts (SBP)-Property J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

42 Boundary conditions: SBP For SBP derivatives there is a norm, such that 2 Telescoping sum is broken at boundary Skew Symmetry only broken at D, and D N,N Boundary flux of our scheme depends entirely on boundary values, addition flux to enforce BC Enforcing BC e.g. in Carpenter 3 give global energy estimates. (Work in progress... ) 2 Strand, JCP 0, p47, Carpenter et al, JCP 08, p272, 993 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

43 Skew Symmetric discretisation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

44 Skew Symmetric discretisation Without SBP J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

45 Skew Symmetric discretisation With SBP J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

46 Skew Symmetric discretisation Conservation with boundary J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

47 Skew Symmetric discretisation First Conclusions Skew symmetry leads to kinetic energy conservation... with telescoping sum property full conservation strict skew symmetry & perfect conservation on transformed grids Procedure easy to implement & numerically efficient Derivatives of any order in space Time stepping of any order J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

48 Skew Symmetric discretisation First Conclusions Skew symmetry leads to kinetic energy conservation... with telescoping sum property full conservation strict skew symmetry & perfect conservation on transformed grids Procedure easy to implement & numerically efficient Derivatives of any order in space Time stepping of any order Outlook Implementation for shock acoustic simulations in progress Big DNS of turbulent flows J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

49 Bonus Part Schemes for the incompressible Navier-Stokes equation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

50 Bonus Part Schemes for the incompressible Navier-Stokes equation How to avoid odd-even decoupling? Idea: energy and momentum conserving, and collocated, by use of skew symmetry & the combination of non-symmetric derivatives Grid transformations preserving these properties general in 2D restricted in 3D J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

51 Incompressible Navier-Stokes t ρ+ xα ρu α = 0 ρ t ( ρu α ) + ( xβ ρu β +ρu β xβ )u α + xα p 2 = β τ α,β γ tp + γ γ x β (u β p) u β xβ p = 0 t u α + ( xβ u β +u β xβ ) u α + xα p 2 = ν u α xα u α = 0 α, β =, 2, 3 Pressure Poission Equation: xα xα p = ( xα D u ) u α + ν u α J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

52 Incompressible Navier-Stokes t u α + ( xβ u β +u β xβ ) u α + xα p 2 = ν u α xα u α = 0 α, β =, 2, 3 Pressure Poission Equation: xα xα p = ( xα D u ) u α + ν u α J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

53 Incompressible Navier-Stokes t u α + 2 Pressure Poission Equation: Discrete ( xβ u β +u β xβ ) u α + xα p = ν u α xα xα p = xα ( D u u α + ν u α ) xα xα p D α G α p xα u α = 0 α, β =, 2, 3 (Gp) i (p i+ p i ) & (Du) i (u i+ u i ) }{{} DGp (p i+2 2p i + p i 2 ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

54 Incompressible Navier-Stokes t u α + 2 Pressure Poission Equation: Discrete ( xβ u β +u β xβ ) u α + xα p = ν u α xα xα p = xα ( D u u α + ν u α ) xα xα p D α G α p xα u α = 0 α, β =, 2, 3 (Gp) i (p i+ p i ) & (Du) i (u i+ u i ) }{{} DGp (p i+2 2p i + p i 2 ) Odd-even decoupling, nasty to solve! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

55 Avoiding decoupling of p Staggered Grid Collocated Grid u i-/2,j v i,j+/2 p i,j u i+/2,j v i,j p i,j u i,j v i,j-/2 + Simple + No extra damping + Nice Boundary + Same geometry factors - Boundary non-simple - Different geometry factors for p, u α - Upwind: Large damping - Rhie-Chow: Smaller Damping J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

56 Avoiding decoupling of p Staggered Grid Collocated Grid u i-/2,j v i,j+/2 p i,j u i+/2,j v i,j p i,j u i,j v i,j-/2 + Simple + No extra damping + Nice Boundary + Same geometry factors - Boundary non-simple - Different geometry factors for p, u α - Upwind: Large damping - Rhie-Chow: Smaller Damping Skew Symmetric Schemes J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

57 Avoiding decoupling of p Staggered Grid Collocated Grid u i-/2,j v i,j+/2 p i,j u i+/2,j v i,j p i,j u i,j v i,j-/2 + Simple + No extra damping + Nice Boundary + Same geometry factors - Boundary non-simple - Different geometry factors for p, u α - Upwind: Large damping - Rhie-Chow: Smaller Damping Skew Symmetric Schemes??? J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

58 Discretisation t u α + 2 xα u α = 0 ( xβ u β +u β xβ ) u α + xα p = ν u α D α u α = 0 t u α + 2 (D βu β + U β G β ) u α + G α p }{{} = νlu α D u Transport term is skew symmetric D ut = 2 (U βd T β + GT β U β) D u provided D β = G T β not necessary skew sym.! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

59 Discrete Poission Equation D α u α = 0 t u α + D u u α + G α p = νlu α D α G α p = D α ( D u u α + νlu α ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

60 Discrete Poission Equation D α u α = 0 t u α + D u u α + G α p = νlu α D α G α p = D α ( D u u α + νlu α ) D β = G T β }{{} Energy cons.! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

61 Discrete Poission Equation D β = G T β }{{} Energy cons.! = h D α u α = 0 t u α + D u u α + G α p = νlu α D α G α p = D α ( D u u α + νlu α ) Not upwind! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

62 Discrete Poission Equation D β = G T β }{{} Energy cons.! = h Not upwind! D α u α = 0 t u α + D u u α + G α p = νlu α D α G α p = D α ( D u u α + νlu α ) D β G β = h J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

63 Discrete Poission Equation D β = G T β }{{} Energy cons.! = h Not upwind! D α u α = 0 t u α + D u u α + G α p = νlu α D α G α p = D α ( D u u α + νlu α ) D β = (... D β G β = h /6 /2 /3 0 ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

64 Numerical example J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

65 Numerical example J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

66 Numerical example J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

67 Transformed Grids D α u α = 0 t u α + 2 (D βu β + U β G β ) u α + G α p = νlu α J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

68 Transformed Grids D α u α = 0 t u α + 2 (D βu β + U β G β ) u α + G α p = νlu α D β M α,β u α = 0 J t u α + 2 ( Dγ M β,γ U β + U β M β,γ Ḡ γ ) uα + M α,γ Ḡ γ p = νlu α D β = ḠT β J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

69 Transformed Grids D α u α = 0 t u α + 2 (D βu β + U β G β ) u α + G α p = νlu α D β M α,β u α = 0 J t u α + 2 ( Dγ M β,γ U β + U β M β,γ Ḡ γ ) uα + M α,γ Ḡ γ p = νlu α D β = ḠT β p D M α,β M α,γ β Ḡ γ p =... J J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

70 Transformed Grids: Conservation? D β M α,β u α = 0 J t u α + 2 ( Dγ M β,γ U β + U β M β,γ Ḡ γ ) uα + M α,γ Ḡ γ p = νlu α D β = ḠT β J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

71 Transformed Grids: Conservation? D β M α,β u α = 0 J t u α + 2 ( Dγ M β,γ U β + U β M β,γ Ḡ γ ) uα + M α,γ Ḡ γ p = νlu α D β = ḠT β Energy conservation uα T M α,γ Ḡ γ p = p T Ḡγ T M α,γ u α = p T Dγ M α,γ u α = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

72 Transformed Grids: Conservation? D β M α,β u α = 0 J t u α + 2 ( Dγ M β,γ U β + U β M β,γ Ḡ γ ) uα + M α,γ Ḡ γ p = νlu α D β = ḠT β Energy conservation Momentum conservation uα T M α,γ Ḡ γ p = p T Ḡγ T M α,γ u α = p T Dγ M α,γ u α = 0 T M α,γ Ḡ γ p = p T Ḡγ T m α,γ = p T Dγ m α,γ =? J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

73 Transformed Grids: Momentum Conservation J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

74 Transformed Grids: Momentum Conservation 2D D γ m α,γ = ( Dξ y η D η y ξ... Provided y ξ = D ξ y y η = D η y ) = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

75 Transformed Grids: Momentum Conservation 2D D γ m α,γ = ( Dξ y η D η y ξ... Provided y ξ = D ξ y y η = D η y ) = 0 3D D T γ m α,γ = ( Dξ (y η z ζ y ζ z η ) + D η (y ζ z ξ y ξ z ζ ) + D ζ (y ξ z η y η z ξ )... Only zero for restricted transformations: x(ξ, η), y(ξ, η), z(ζ) ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

76 Second Conclusions We found Schemes for the incompressible Navier-Stokes equation, wich are collocated, energy and momentum conserving, by use of skew symmetry & the combination of non-symmetric derivatives Grid transformations preserving these properties general in 2D restricted in 3D Very simple to implement J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

77 Conclusions Skew symmetric Schemes are Skew symmetric schemes respect kinetic energy Avoid numerical damping Stable Simple to implement... worth a try! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

78 Thank you for your attention! Question? J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

79 Euler Equations, more dimensions ρ t ( ρu α ) + 2 [ xβ ρu β +ρu β xβ ]u α + xα p = 0. Momentum equation in skew symmetric form, 2D, u u component : J ρ t ( ρu) + [ ( ξ ϱ(y η u x η v) +ϱ(y η u x η v) ξ ) 2 ] +( η ϱ( y ξ u + x ξ v) +ϱ( y ξ u + x ξ v) η ) u +( ξ y η η y ξ )p = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

80 Shock & Artifical Damping 2 ( tr + R t )u + (DUR + RUD)u + Dp = Dτ 2 Friction Term τ = µdu DµDu FσF T u with adaptive sigama: σ = ( r th + 2 r r ) th i with with shock detector building on dilatation r i = r i ( u) r i 4 Bogey et al, JCP 228, 447, 2009 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

81 Where are we now? Shock & Acousic Pulse ϱ J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

82 Skew Symmetric discretisation Euler Equations, time t ϱ + B u ϱ = 0 2 [ tr + R t ]u + 2 Duϱ u + D x p = 0 γ tp + γ γ Bu p C u p = 0 Leap Frog like scheme ( ϱ n+ ϱ n ) +B un ϱ n = 0 4 t 2 t ( (R n+ + R n )u n+ (R n + R n )u n ) + 2 Dun ϱ n u n + D x p n = 0 γ 2 t ( p n+ p n ) +γ Bun p n γ Cun p n = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

83 Skew Symmetric discretisation Euler Equations, implicit time mass (ϱ n+ ϱ n ) + ( 2 t 8 D (uϱ) n + 6(uϱ) n + (uϱ) n+) = 0 velocity ( (R n+ + R n )u n+ (R n + R n )u n ) + 4 t (D (uϱ)n u n + D (uϱ)n (u n + 4u n + u n+ ) + D (uϱ)n+ u n+) D x(p n + 2p n + p n+ ) = 0 pressure 0 = ( p n+ p n ) + γ 2 t γ ( 4γ D (up) n + 2(up) n + (up) n+) 4 un D(p n + 2p n + p n+ ) J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

84 Time Integration Energy conservation Time integration by midpoint rule produces energy conservation u n+ u n t + 3 Dun+/2 u n+/2 = 0 Multiplying by (u n+/2 ) T = 2 (un + u n+ ) T 2 (un + u n+ ) T un+ u n + t 3 (un+/2 ) T D un+/2 u n+/2 = 0 }{{} =0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

85 Time Integration Energy conservation Time integration by midpoint rule produces energy conservation u n+ u n t + 3 Dun+/2 u n+/2 = 0 Multiplying by (u n+/2 ) T = 2 (un + u n+ ) T 2 (un + u n+ ) T un+ u n t = 0 gives ( (u n+ ) T u n+ (u n ) T u n) = 0 2 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

86 Time Integration Momentum conservation Time integration by midpoint rule produces momentum conservation Multiplying by ( ) T = (,,,,,... ) ( ) T un+ u n t + 3 ( ) T D un+/2 u n+/2 = 0 with ( ) T D un+/2 u n+/2 = ( ) T DU n+/2 + (u n+/2 ) T Du n+/2 = 0 gives ( u n+ 2 i i i u n i ) = 0 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

87 Euler Equations, more dimensions Use divergence as 2 ( tϱ + ϱ t ) u i + [ ϱ u + ϱ u ] u i + p = 0. 2 u = J ξ i (e j e k )u = (e j e k ) J ξ i u. i,cy Momentum equation in skew symmetric form 2 ( tϱ + ϱ t ) u α + ( ) ξ i (e j e k )ϱu + (e j e k )ϱu 2 J ξ i uα i,cy + J ξ i (e j e k )p = 0 i,cy... is indeed skew symmetric! J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46 α i,cy

88 Time discretization Central scheme 3 step 4 t t ϱ + RHS ϱ = 0 2 [ tr + R t ]u + RHS u = 0 γ tp + RHS p = 0 2 t ( ϱ n+ ϱ n ) +RHS ϱ = 0 ( (R n+ + R n )u n+ (R n + R n )u n ) +RHS u = 0 ( p n+ p n ) +RHS p = 0 γ 2 t J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

89 Skew Symmetric discretisation Euler Equations, time mass n+ 2 = T (ϱn + ϱ n+ ) 2 mom n+ 2 = (ϱ n + ϱ n+ ) T 2 (u n + u n+ ) 2 e n+/2 tot = T γ (pn + p n+ )/2 + (un ) T (ϱ n + ϱ n+ )u n+ 4 E.g. momentum conservation mom /2 mom N /2 = equivalent to FV for t 0 N n= N f /2 n= f N /2 J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

90 End J. Reiss (TU Berlin) Skew-Symmetric Schemes August 8 th, / 46

Conservative, skew symmetric compact finite difference schemes for the compessible Navier Stokes equations

Conservative, skew symmetric compact finite difference schemes for the compessible Navier Stokes equations Jan Schulze, Julius Reiss & Jörn Sesterhenn Fachgebiet für Numerische Fluidmechanik TU Berlin