Failsafe FCT algorithms for strongly time-dependent flows with application to an idealized Z-pinch implosion model

Transcription

1 Failsafe FCT algorithms for strongly time-dependent flows with application to an idealized Z-pinch implosion model Matthias Möller 1 joint work with: Dmitri Kuzmin 2 1 Institute of Applied Mathematics (LS3), Dortmund University of Technology, Germany, matthias.moeller@math.tu-dortmund.de 2 Chair of Applied Mathematics III, University Erlangen-Nuremberg, Germany kuzmin@am.uni-erlangen.de

2 Motivation: Z-pinch implosion Phenomenological model by Banks and Shadid compressible Euler equations + source term coupled with tracer equation for Lorentz force Mathematical challenges high-resolution scheme for time-dependent conservation laws; positivity-preservation of density and pressure; failsafe strategy

3 High-resolution schemes for t U + A U = 0 High-order scheme M C du H dt = KU H Low-order scheme M L du L dt = LU L, L = K + D Predictor: Compute low-order solution M L du L dt = LU L U L M 1 L LU L Corrector: Apply limited antidiffusion M L U = M L U L + F, F = [M L M C ] U L DU L

4 High-resolution schemes for t U + A U = 0 High-order scheme M C du H dt = KU H Low-order scheme M L du L dt = LU L, L = K + D Predictor: Compute low-order solution M L du L dt = LU L U L M 1 L LU L Corrector: Apply limited antidiffusion M L U = M L U L + F, F = [M L M C ] U L DU L Low-order scheme must satisfy physical constraints

5 Design principles of FCT schemes Perform flux correction such that mass is conserved.

6 Design principles of FCT schemes Perform flux correction such that mass is conserved. Conservative flux decomposition m i U i = m i U L i + j i F ij, F ji = F ij

7 Design principles of FCT schemes Perform flux correction such that mass is conserved. Conservative flux decomposition and limiting m i U i = m i U L i + j i α ij F ij, F ji = F ij, α ji = α ij high-order approximation (α ij = 1) to be used in smooth regions low-order approximation (α ij = 0) to be used near steep fronts

8 Design principles of FCT schemes, cont d Perform flux correction such that certain physical quantities are bounded by the local extrema of the low-order solution.

9 Design principles of FCT schemes, cont d Perform flux correction such that certain physical quantities are bounded by the local extrema of the low-order solution. u min i u L i + 1 m i j i α u ij f u ij umax i u L i

10 Fixed fraction flux limiter Set α (0) ij := 1 and repeat r = 1,..., R Mark all nodes i that violate the local FCT constraint u min i u L i + 1 m i j i α (r 1) ij fij u umax i Eliminate fixed fraction of unacceptable antidiffusion { 1 r/r if node i or j is marked α (r) ij := otherwise α (r 1) ij Low-order solution is recovered in the worst case: U i = U L i

11 Zalesak s flux limiter Consider positive/negative antidiffusive contributions separately P + i u i P i

12 Zalesak s flux limiter Q + i P + i Q i P i Consider positive/negative antidiffusive contributions separately Limit antidiffusion if it exceeds the distance to local maximum/minimum

13 Zalesak s flux limiter Q + i P + i Q i P i Consider positive/negative antidiffusive contributions separately Limit antidiffusion if it exceeds the distance to local maximum/minimum Nodal correction factors R + i = min{1, Q + i /P + i } for positive fluxes into node i R i = min{1, Q i /P i } for negative fluxes into node i

14 Zalesak s flux limiter Q + i P + i Q i P i Consider positive/negative antidiffusive contributions separately Limit antidiffusion if it exceeds the distance to local maximum/minimum Nodal correction factors R + i = min{1, Q + i /P + i R i = min{1, Q i /P i } for positive fluxes into node i } for negative fluxes into node i Limit antidiffusive flux for edge ij by the minimum of R i and R j

15 Flux limiting for systems Apply flux limiter to a set of control variables, e.g., the primitive variables density, pressure and velocity.

16 Flux limiting for systems Apply flux limiter to a set of control variables, e.g., the primitive variables density, pressure and velocity. f ρ ij f p ij f v ij FCT FCT FCT min{α ρ ij, αp ij, αv ij}

17 Flux limiting for systems Apply flux limiter to a set of control variables, e.g., the primitive variables density, pressure and velocity. f ρ ij f p ij f v ij f ρ ij α ρ ij f p ij α p ij αρ ij f v ij FCT FCT FCT or FCT FCT FCT min{α ρ ij, αp ij, αv ij} α ρ ij α p ij αρ ij α v ijα p ij αρ ij

18 Flux limiting for systems Apply flux limiter to a set of control variables, e.g., the primitive variables density, pressure and velocity. f ρ ij f p ij f v ij f ρ ij α ρ ij f p ij α p ij αρ ij f v ij FCT FCT FCT or FCT FCT FCT min{α ρ ij, αp ij, αv ij} α ρ ij α p ij αρ ij α v ijα p ij αρ ij Nodal transformation of variables V i = T (U i )U i and G ij = T (U i )F ij

19 Failsafe flux correction algorithm 1 Compute low-order solution at time t n+1 M L U L U n t = θlu L + (1 θ)lu n prediction 2 Perform flux correction by Zalesak s limiter m i U (0) i = m i U L i + j i α ij F ij correction 3 Eliminate spurious undershoots/overshoots m i U (r) i = m i Ui L + α (r) ij [α ijf ij ] j i failsafe step

20 Double Mach reflection From: P.R. Woodward and P. Colella, JCP 54, 115 (1984) U L = cos sin Γ L Γ R U R = Γ out Γ wall

21 Double Mach reflection, cont d Solution at T = 0.2 computed by low-order scheme (α ij 0) density distribution density distribution h = 1/64 t = h = 1/128 t =

22 Double Mach reflection, cont d Solution at T = 0.2 computed by fixed fraction flux limiter density distribution density distribution h = 1/64 t = h = 1/128 t =

23 Double Mach reflection, cont d Solution at T = 0.2 computed by Zalesak s flux limiter (α ij = α p ij αρ ij ) density distribution density distribution h = 1/64 t = h = 1/128 t =

24 Double Mach reflection, cont d Solution at T = 0.2 computed by Zalesak s flux limiter (α ij = α p ij αρ ij ) density distribution pressure distribution h = 1/256 t = h = 1/256 t = Numerical studies: D. Kuzmin, M. Shashkov, J. Shadid, M.M. (to appear in JCP)

25 Idealized Z-pinch implosion model by Banks and Shadid Generalized Euler system coupled with scalar tracer equation ρ ρv 0 ρv t ρe + ρv v + pi ρev + pv = f f v ρλ ρλv 0 Equation of state p = (γ 1)ρ(E 0.5 v 2 ) Non-dimensional Lorentz force ( ) 2 f = (ρλ) I(t) I max êr r eff

26 Coupled solution algorithm time stepping loop Given: (U, ρλ) n Low-order scheme (ρλ) L,(k) M L du dt = L(U)U + S(v, ρλ) M L d(ρλ) dt = L(v)(ρλ) v L,(k) (U, ρλ) L Zalesak s limiter α ij = α ρλ ij αp ij αρ ij [+ failsafe corr.] (U, ρλ) n+1

27 Idealized Z-pinch implosion From: J.W. Banks and J.N. Shadid, JCP 61, 725 (2009) ρ = 1.0 λ = 0.0 ρ = 10 6 λ = 1.0 = 0.05 ρ = 0.5 λ = 0.0 v = 0.0, p = 1.0 everywhere

28 Idealized Z-pinch implosion, cont d density distribution

29 Idealized Z-pinch implosion, cont d density distribution density distribution

30 Idealized Z-pinch implosion, cont d density distribution density distribution density distribution

31 Idealized Z-pinch implosion, cont d radius exact R(t)=1 t 4 low order Rusanov FCT lumped mass FCT consistent mass time shell radius as a function of time z density symmetry plot, t = 0.9 density symmetry plot, t = 1.05 R(t) r

32 Idealized Z-pinch implosion, cont d Initialization radius exact R(t)=1 t 4 low order Rusanov FCT lumped mass FCT consistent mass time shell radius as a function of time z density symmetry plot, t = 0.9 density symmetry plot, t = 1.05 R(t) r

33 Conclusions and outlook Linearized flux correction algorithm for time-dependent flows mass conservation, boundedness of physical quantities, failsafe strategy if density/pressure becomes negative Coupled solution algorithm for idealized Z-pinch implosions positivity and symmetry preservation on unstructured grids Todo: comparative study and extension to realistic scenarios current drive, r-z plane, RT-instabilities, AMR

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35 Appendix

36 Extended version of Zalesak s FCT limiter Return Input: auxiliary solution u L and antidiffusive fluxes f u ij, where f u ji f u ij 1 Sums of positive/negative antidiffusive fluxes into node i P + i = j i max{0, f u ij}, P i = j i min{0, f u ij} 2 Upper/lower bounds based on the local extrema of u L Q + i = m i (u max i u L i ), Q i = m i (u min i u L i ) 3 Correction factors αij u = αu ji to satisfy the FCT constraints α u ij = min{r ij, R ji }, R ij = { min{1, Q + i /P + i } if f u ij 0 min{1, Q i /P i } if f u ij < 0

37 Node-based transformation of control variables Return Conservative variables: density, momentum, total energy [ ] U i = [ρ i, (ρv) i, (ρe) i ], F ij = f ρ ij, f ρv ij, f ρe ij, F ji = F ij Primitive variables V = T U: density, velocity, pressure ] V i = [ρ i, v i, p i ], v i = (ρv)i ρ i, p i = (γ 1) [(ρe) i (ρv)i 2 2ρ i G ij = [ f ρ ij, f v ij, f p ij] = T (Ui )F ij, T (U j )F ji = G ji G ij Raw antidiffusive fluxes for the velocity and pressure [ fij v = f ρv ij vif ρ ij ρ i, f p ij = (γ 1) vi 2 2 f ρ ij v i f ρv ij ] + f ρe ij

38 Constrained initialization Pointwise initialization U(x i ) = U 0 (x i ) { 1.0 in Ω1 ρ = 0.01 in Ω 2 u = v = 0.0, p = 1.0

39 Constrained initialization Pointwise initialization U(x i ) = U 0 (x i ) Conservative initialization Ω wu h dx = Ω wu 0 dx { 1.0 in Ω1 ρ = 0.01 in Ω 2 u = v = 0.0, p = 1.0

40 Constrained initialization Pointwise initialization U(x i ) = U 0 (x i ) Conservative initialization Ω wu h dx = Ω wu 0 dx Consistent L 2 -projection j m ijuj H = Ω ϕ iu 0 dx { 1.0 in Ω1 ρ = 0.01 in Ω 2 u = v = 0.0, p = 1.0

41 Constrained initialization Pointwise initialization U(x i ) = U 0 (x i ) Conservative initialization Ω wu h dx = Ω wu 0 dx Consistent L 2 -projection j m ijuj H = Ω ϕ iu 0 dx Mass-lumped L 2 -projection m i U L i = Ω ϕ iu 0 dx { 1.0 in Ω1 ρ = 0.01 in Ω 2 u = v = 0.0, p = 1.0

42 Constrained initialization Pointwise initialization U(x i ) = U 0 (x i ) Conservative initialization Ω wu h dx = Ω wu 0 dx Consistent L 2 -projection j m ijuj H = Ω ϕ iu 0 dx Mass-lumped L 2 -projection m i U L i = Ω ϕ iu 0 dx { 1.0 in Ω1 ρ = 0.01 in Ω 2 u = v = 0.0, p = 1.0 Limited L 2 -projection (0 α ij 1) m i U i = m i U L i + j i α ij m ij (U L i U L j )

43 Initialization for bilinear elements (a) consistent L 2-projection (b) lumped L 2-projection (c) L 2-projection, α ij = α ρ ij bilinear elements, 3 3 Gauss rule ρ ρ h 2 min(ρ h ) max(ρ h ) (a) 1.048e e (b) 1.168e e (c) 1.103e e computed by adaptive cubature formulae

44 Initialization for linear elements Conclusions (a) consistent L 2-projection (b) lumped L 2-projection (c) L 2-projection, α ij = α ρ ij linear elements, 3-point Gauss rule ρ ρ h 2 min(ρ h ) max(ρ h ) (a) 1.206e e (b) 1.357e e (c) 1.259e e computed by adaptive cubature formulae