Numerical methods for eigenvalue problems in the description of drift instabilities in the plasma edge

Transcription

1 Numerical methods for eigenvalue problems in the description of drift instabilities in the plasma edge Dominik Löchel Supervisors: M. Hochbruck and M. Tokar Graduate school Dynamic of hot plasmas Mathematisches Institut Heinrich-Heine-Universität Düsseldorf April 2009

2 Outline Physical background Numerical methods Summary and perspectives

3 Energy gain by nuclear fusion fusion of deuterium and tritium to helium overcoming the Coulomb barrier high temperature and high density for a long time plasma magnetic confinement

4 Tokamak toroidal coil

5 Tokamak toroidal coil magnetic field lines within the flux surface

6 Tokamak toroidal coil magnetic field lines within the flux surface

7 Tokamak v E B toroidal coil magnetic field lines within the flux surface

8 Tokamak toroidal coil magnetic field lines within the flux surface primary coil transformer iron

9 Drift instabilities

10 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r + ik y iωt), I(ω) maximal

11 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

12 Drift instabilities φ(θ, t) = φ(θ) exp(ikr r +ik y iωt) 1 φ R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

13 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

14 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

15 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

16 Drift instabilities φ(θ, t) = φ(θ) exp(ikr r +ik y iωt) 1 φ R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

17 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ

18 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) φ 2 R(φ) I(φ) 0 HFS θ LFS perturbation wave φ anomalous transport Γ

19 Drift instabilities φ(θ, t) = φ(θ) exp(ik r r +ik y iωt) 1 φ 2 T / ev T n p = T (θ) n(θ) n / m R(φ) I(φ) HFS θ LFS perturbation wave φ 0 HFS θ LFS 0 anomalous transport Γ

20 Anomalous transport model transport equations: particle continuity momentum balance Faraday s law Amperè s law Ohm s law zero current divergence

21 Anomalous transport model transport equations: particle continuity momentum balance Faraday s law Amperè s law Ohm s law zero current divergence separation f = f0 }{{} macroscopical + f }{{} microscopical, perturbation

22 Anomalous transport model transport equations: particle continuity momentum balance Faraday s law Amperè s law Ohm s law zero current divergence separation f = f0 }{{} macroscopical linearization of equations and applying + f }{{} microscopical, perturbation f (θ, t 0) = f (θ) exp(ikr r + ik y iωt)

23 Eigenvalue equation 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3

24 Eigenvalue equation generic form: 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 a j, b j, φ: [0, 2π[ C, 2π-periodic smooth functions in θ

25 Eigenvalue equation generic form: 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 a j, b j, φ: [0, 2π[ C, 2π-periodic smooth functions in θ desired: eigenpair (ω, φ) where the growth rate I(ω) is maximal

26 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0

27 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 Discretize equation on grid of N points:

28 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 Discretize equation on grid of N points: [0, 2π[ θ

29 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 Discretize equation on grid of N points: [0, 2π[ θ a(θ) Diag(a( θ))

30 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 Discretize equation on grid of N points: [0, 2π[ θ a(θ) Diag(a( θ)) 2 D θ 2 2 finite differences or pseudo-spectral-method k f m θ k (θ 0) ψ j, f k ψ j θ k (θ 0) j=0

31 Discretization [ ω 3 a 3 + ω 2 a 2 + ω (a 2 ) 2 ] 1 + b 1 θ 2 + a 0 + b 0 θ 2 φ = 0 Discretize equation on grid of N points: [0, 2π[ θ a(θ) Diag(a( θ)) 2 D θ 2 2 finite differences or pseudo-spectral-method k f m θ k (θ 0) ψ j, f k ψ j θ k (θ 0) j=0 P(ω) φ := ( ω 3 M 3 + ω 2 M 2 + ωm 1 + M 0 ) φ = 0

32 Polynomial eigenvalue problem ( ω 3 M 3 + ω 2 M 2 + ωm 1 + M 0 ) φ = 0 3 ω M I M 2 M 1 M 0 + I ω2 φ ω φ = 0. I I φ ωbx = Ax, (generalized eigenvalue problem)

33 Polynomial eigenvalue problem ( ω 3 M 3 + ω 2 M 2 + ωm 1 + M 0 ) φ = 0 3 ω M I M 2 M 1 M 0 + I ω2 φ ω φ = 0. I I φ ωbx = Ax, (generalized eigenvalue problem) N = QZ -algorithm: 1 hour.

34 Polynomial eigenvalue problem ( ω 3 M 3 + ω 2 M 2 + ωm 1 + M 0 ) φ = 0 3 ω M I M 2 M 1 M 0 + I ω2 φ ω φ = 0. I I φ ωbx = Ax, (generalized eigenvalue problem) N = QZ -algorithm: 1 hour. simulations with eigenvalue equations: 10 years

35 Polynomial eigenvalue problem ( ω 3 M 3 + ω 2 M 2 + ωm 1 + M 0 ) φ = 0 3 ω M I M 2 M 1 M 0 + I ω2 φ ω φ = 0. I I φ ωbx = Ax, (generalized eigenvalue problem) N = QZ -algorithm: 1 hour. simulations with eigenvalue equations: 10 years only one eigenpair desired iterative solver, especially Jacobi-Davidson algorithm

36 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N

37 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V

38 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0

39 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y).

40 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u.

41 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u. if r small enough do Stop end if

42 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u. if r small enough do Stop end if (5.) solve (approximately) (e.g. GMRES) (I w u H u H w ) P(ν)(I u u H ) t = r, t u, w := P (ν) u. (6.) expand search space to [V, t]. end loop

43 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u. if r small enough do Stop end if (5.) solve (approximately) (e.g. GMRES) (I w u H u H w ) P(ν)(I u u H ) t = r, t u, w := P (ν) u. (6.) expand search space to [V, t]. end loop

44 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u. if r small enough do Stop end if (5.) solve (approximately) (e.g. GMRES) (I w u H u H w ) P(ν)(I u u H ) t = r, t u, w := P (ν) u. (6.) expand search space to [V, t]. end loop

45 Jacobi-Davidson algorithm (0.) choose search space V C N k, k N loop (1.) orthonormalize V (2.) calculate eigenpairs (ν, y) of V H P(ν)V y = 0 (3.) select Ritz pair (ν, u := V y). (4.) calculate residual r := P(ν) u. if r small enough do Stop end if (5.) solve (approximately) (e.g. GMRES) (I w u H u H w ) P(ν)(I u u H ) t = r, t u, w := P (ν) u. (6.) expand search space to [V, t]. end loop

46 Jacobi-Davidson approach (5.) Solve (approximately) w u (I H ) u H P(ν)(I u u H ) t = r, w t u

47 Jacobi-Davidson approach (5.) Solve (approximately) w u (I H ) u H P(ν)(I u u H ) t = r, w t u Expansion of search space by one Newton step ( ) P(λ) x F (λ, x) := x H = 0. x 1

48 Jacobi-Davidson approach (5.) Solve (approximately) w u (I H ) u H P(ν)(I u u H ) t = r, w t u Expansion of search space by one Newton step ( ) P(λ) x F (λ, x) := x H = 0. x 1 Another decomposition: one step approximation t = u H Q(ν) r u H Q(ν)P (ν) u Q(ν)P (ν) u Q(ν) r, Q(ν)P (ν) = I Q(ν) r = z P (ν) z = r, Q(ν)P (ν) u = s P (ν) s = P (ν) u P (ν) = =

49 Jacobi-Davidson approach (5.) Solve (approximately) w u (I H ) u H P(ν)(I u u H ) t = r, w t u Expansion of search space by one Newton step ( ) P(λ) x F (λ, x) := x H = 0. x 1 Another decomposition: one step approximation t = u H Q(ν) r u H Q(ν)P (ν) u Q(ν)P (ν) u Q(ν) r, Q(ν)P f (ν) = I Q(ν) r = z P f (ν) z = r, Q(ν)P (ν) u = s P f (ν) s = P (ν) u P f (ν) = =

50 Jacobi-Davidson approach (3.) select Ritz pair (ν, u := V y). Physics: strongest growth rate I(ω). V = [ v 1 ], v 1 random vector

51 Jacobi-Davidson approach (3.) select Ritz pair (ν, u := V y). Physics: strongest growth rate I(ω). V = [ v 1 ], v 1 random vector

52 Jacobi-Davidson approach (3.) select Ritz pair (ν, u := V y). Physics: strongest growth rate I(ω). V = [ v 1 ], v 1 random vector

53 Jacobi-Davidson approach (3.) select Ritz pair (ν, u := V y). Physics: strongest growth rate I(ω). V = [ v 1 ], v 1 random vector desired eigenpair is not found in general.

54 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ijθ) ) j= m,...,m as long as φ is extremely smooth.

55 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ijθ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ.

56 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ij θ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ. Realization: calculation on coarser grid.

57 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ij θ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ. Realization: calculation on coarser grid. Advantages: coarsest grid (e.g. N = 8) QZ, selection of eigenpair.

58 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ij θ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ. Realization: calculation on coarser grid. Advantages: coarsest grid (e.g. N = 8) QZ, selection of eigenpair. high accuracy due to pseudo-spectral method or optimized grid positions.

59 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ij θ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ. Realization: calculation on coarser grid. Advantages: coarsest grid (e.g. N = 8) QZ, selection of eigenpair. high accuracy due to pseudo-spectral method or optimized grid positions. improvement of eigenpair on the next finer grid by Jacobi-Davidson method.

60 Jacobi-Davidson approach (0.) choose search space V C N k, k N Possibilities: random vector selection of Ritz pair doubtful. V = ( exp(ij θ) ) j= m,...,m as long as φ is extremely smooth. Idea: get (cheap) approximation on φ. Realization: calculation on coarser grid. Advantages: coarsest grid (e.g. N = 8) QZ, selection of eigenpair. high accuracy due to pseudo-spectral method or optimized grid positions. improvement of eigenpair on the next finer grid by Jacobi-Davidson method. selection of Ritz pair by similarity to coarse grid approximation.

61 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = 8 0 LFS θ HFS

62 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = 16 0 LFS θ HFS

63 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = 32 0 LFS θ HFS

64 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = 64 0 LFS θ HFS

65 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

66 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

67 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

68 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

69 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

70 Multilevel Jacobi-Davidson approach Example: 1 φ 2 N = LFS θ HFS

71 Multilevel Jacobi-Davidson approach Example: ω ω N dimv time/s

72 Multilevel Jacobi-Davidson approach Example: ω ω N dimv time/s

73 Wave number Wave number K 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3

74 Wave number Wave number K 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 Γ = I C(T, p, K ) I(ω(K )) 3 ω(k ) 2 φ(k ) 2 dk

75 Wave number Wave number K 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 Γ = I C(T, p, K ) I(ω(K )) 3 ω(k ) 2 φ(k ) 2 dk discrete representation of wave number range.

76 Wave number Wave number K 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 Γ = I C(T, p, K ) I(ω(K )) 3 ω(k ) 2 φ(k ) 2 dk discrete representation of wave number range. define modes (ω(k ), φ(k )).

77 Wave number Wave number K 2 φ θ 2 = ω( β + zγ 3 µk 2) (1 + λ) βk + izγ 3 ĈK 2 γ1 (K 3z ω ( ) ) 1 + zγ 3 (1 + α)k 2 ( (1 + α) γ ) 2 L B (1 zγ 3 ωk ) + zω(ω + αk ) φ γ 3 Γ = I C(T, p, K ) I(ω(K )) 3 ω(k ) 2 φ(k ) 2 dk discrete representation of wave number range. define modes (ω(k ), φ(k )). track modes and find wave number of maximal growth rate.

78 Wave number K = 0.4, K = R(φ) HFS θ LFS

79 Wave number K = 0.4, K = R(φ) HFS θ LFS sim(ω, ω) = exp( ω ω ) sim( φ, φ) = φ H φ

80 Wave number K = 0.4, K = R(φ) HFS θ LFS sim(ω, ω) = exp( ω ω ) sim( φ, φ) = φ H φ sim ( (ω, φ), (ω, φ) ) := sim(ω, ω) sim( φ, φ)

81 Wave number K = 0.4, K = R(φ) HFS θ LFS sim(ω, ω) = exp( ω ω ) sim( φ, φ) = φ H φ sim ( (ω, φ), (ω, φ) ) := sim(ω, ω) sim( φ, φ) sim(ω, ω)= , sim( φ, φ)=

82 Wave number K = 0.4, K = R(φ) HFS θ LFS sim(ω, ω) = exp( ω ω ) sim( φ, φ) = φ H φ sim ( (ω, φ), (ω, φ) ) := sim(ω, ω) sim( φ, φ) sim(ω, ω)= , sim( φ, φ)=

83 Wave number K = 0.4, K = R(φ) HFS θ LFS sim(ω, ω) = exp( ω ω ) sim( φ, φ) = φ H φ sim ( (ω, φ), (ω, φ) ) := sim(ω, ω) sim( φ, φ) sim(ω, ω)= , sim( φ, φ)=

84 0.08 Wave number

85 0.08 Wave number I(ω) K

86 0.08 Wave number I(ω) φ HFS θ LFS K

87 Eigenvalue equation: New possibilities before: Mathieu equation with T, n T / ev I(ω) 5 0 HFS θ LFS K Γ / m 2 s Γ = C(T, p, K ) I(ω)3 ω 2 φ 2 0 HFS θ LFS

88 Eigenvalue equation: 0.2 I(ω) New possibilities before: Mathieu equation with T, n now: inhomogeneous profiles (MARFE) T / ev HFS θ LFS K Γ / m 2 s Γ = C(T, p, K ) I(ω)3 ω 2 φ 2 0 HFS θ LFS

89 Eigenvalue equation: 0.2 I(ω) New possibilities before: Mathieu equation with T, n now: inhomogeneous profiles (MARFE) T / ev HFS θ LFS K Γ / m 2 s Γ = C(T, p, K ) I(ω)3 ω 2 φ 2 0 HFS θ LFS

90 Self-consistent calculation eigenvalue equation (T, p)

91 Self-consistent calculation eigenvalue equation (T, p) (ω, φ) anomalous transport

92 Self-consistent calculation eigenvalue equation (T, p) (ω, φ) anomalous transport Γ

93 Self-consistent calculation eigenvalue equation (T, p) (ω, φ) anomalous transport (T new, p new ) Γ global heat balance global particle balance

94 Self-consistent calculation eigenvalue equation (1 α)(t, p) + α(t new, p new ) (ω, φ) damping anomalous transport (T new, p new ) Γ global heat balance global particle balance

95 Self-consistent calculation eigenvalue equation (1 α)(t, p) + α(t new, p new ) (ω, φ) damping anomalous transport (T new, p new ) Γ global heat balance global particle balance damping in the fixed point iteration dynamic damping trust region (T, p), exponential terms (L n, E i )

96 Simulation Self-consistent calculation n /10 19 m temperature profile iterations iterations with evaluations of eigenpairs eigenvalue equations dimv N = N = N = N = N = N = time / minutes 2:42 3:10 1:16 2:55 16:58

97 T / ev Self-consistent calculation neutral particle source at HFS and - - LFS n = m 3 10 n / m HFS θ LFS Γ / m 2 s HFS θ LFS

98 T / ev Self-consistent calculation neutral particle source at HFS and - - LFS n = m 3 n = m 3 10 n / m HFS θ LFS Γ / m 2 s HFS θ LFS

99 T / ev Self-consistent calculation neutral particle source at HFS and - - LFS n = m 3 n = m 3 n = m 3 10 n / m HFS θ LFS Γ / m 2 s HFS θ LFS

100 T / ev Self-consistent calculation neutral particle source at HFS and - - LFS n = m 3 n = m 3 n = m 3 n = m 3 n / m HFS θ LFS Γ / m 2 s HFS θ LFS

101 Summary and perspectives multilevel Jacobi-Davidson eigenvalue solver approximation of search space on coarser grid correction equation by low cost LU decomposition tracking eingenpairs in respect to the wave number self-consistent calculation: fixed point iteration with problem optimized damping strategy transport model needs improvement, especially modeling of heat power from plasma core ( 2d model).