Adaptive Mesh Refinement

Transcription

1 Aleander Knebe, Universidad Autonoma de Madrid Adaptive Mesh Refinement

2 AMR codes

3 Poisson s equation ΔΦ( ) = 4πGρ( )

4 Poisson s equation F ( ) = m Φ( ) ΔΦ( ) = 4πGρ( ) particle approach F ( Gm i ) = i m j ( i j ) ( 3 i j ) i j grid approach ΔΦ( i, j,k ) = 4πGρ( i, j,k ) F ( i, j,k ) = m Φ( i, j,k ) ( i, j,k =position of centre of grid cell (i, j,k))

5 Poisson s equation weapon of choice: tree codes F ( ) = m Φ( ) ΔΦ( ) = 4πGρ( ) particle approach F ( Gm i ) = i m j ( i j ) ( 3 i j ) i j grid approach ΔΦ( i, j,k ) = 4πGρ( i, j,k ) F ( i, j,k ) = m Φ( i, j,k ) ( i, j,k =position of centre of grid cell (i, j,k))

6 Poisson s equation F ( ) = m Φ( ) ΔΦ( ) = 4πGρ( ) particle approach F ( Gm i ) = i m j ( i j ) ( 3 i j ) i j grid approach ΔΦ( i, j,k ) = 4πGρ( i, j,k ) F ( i, j,k ) = m Φ( i, j,k ) ( i, j,k =position of centre of grid cell (i, j,k)) weapon of choice: AMR codes

7 Particle-Mesh Method Particle-Mesh (PM) method ΔΦ( g k,l,m ) = 4πGρ( g k,l,m ) F ( g k,l,m ) = m Φ( g k,l,m ) 1. calculate mass density on grid 2. solve Poisson s equation on grid 3. differentiate potential to get forces 4. interpolate forces back to particles i ρ( g k,l,m ) Φ( g k,l,m ) F ( g k,l,m ) F ( g k,l,m ) F ( i )

8 Particle-Mesh Method numerically integrate Poisson s equation pure PM calculation

9 Particle-Mesh Method numerically integrate Poisson s equation density assignment schemes: interplay pure PM between calculation N and ε: Nε 3 =const. NGP = too crude CIC = common choice TSC = pretty smooth

10 Particle-Mesh Method numerically integrate Poisson s equation pure PM calculation overcoming the spatial resolution limitation?

11 Particle-Mesh Method numerically integrate Poisson s equation AMR calculation Yahagi & Yoshi (2001)

12 Particle-Mesh Method numerically integrate Poisson s equation AMR calculation...details now Yahagi & Yoshi (2001)

13 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

14 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

15 Mesh Refinements types of mesh refinement r refinement: move or stretch the mesh p refinement: adjust the order of the method h refinement: change the mesh spacing

16 Mesh Refinements types of mesh refinement r refinement non-uniform mesh (refined region is known) = advantages: - simple to implement = disadvantages: - difference epression for non-constant zone spacing COSMOS code (Ricker 2000)

17 Mesh Refinements types of mesh refinement r refinement Lagragian mesh (mesh is tied to fluid) = advantages: - constant mass resolution - sharp resolution of contacts = disadvantages: - grid stretching causes numerical dissipation - grid tangling in rotational flows MMH code (Pen 1998)

18 Mesh Refinements types of mesh refinement r refinement Lagragian mesh (mesh is tied to fluid) = advantages: - constant mass resolution - sharp resolution of contacts = disadvantages: - grid stretching causes numerical dissipation - grid tangling in rotational flows usually used only in 1D (e.g. stellar evolution codes) MMH code (Pen 1998)

19 AMR Codes Mesh Refinements types of mesh refinement r refinement arbitrary Lagrangian-Eulerion mesh (mesh moves arbitrarily fluid) = advantages: - Lagrangian mesh where flow is irrotational - Eulerian where mesh distortion is problematic = disadvantages: - difficult to handle... DJEHUTY code (Dearborn et al. 2002)

20 Mesh Refinements types of mesh refinement p refinement not in this course...

21 Mesh Refinements types of mesh refinement h refinement nested grids (static meshes with different resolutions) = advantages: - easy to handle boundaries between meshes = disadvantages: - refined region should not move

22 Mesh Refinements types of mesh refinement h refinement adaptive mesh refinement (refined patches are created and destroyed as needed) = advantages: - fully fleible to problem = disadvantages: - serious book-keeping for grid hierarchy density field of simulated galay cluster AMIGA code (Doumler & Knebe 2010) adaptive grid hierarchy

23 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

24 Adaptive Mesh Refinement adaptive mesh refinement improvements using finer grids double pancake test (Doumler & Knebe 2010)

25 Adaptive Mesh Refinement adaptive mesh refinement improvements using finer grids shock tube test (Teyssier 2002)

26 Adaptive Mesh Refinement adaptive mesh refinement improvements using finer grids grid level shock tube test (Teyssier 2002)

27 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density truncation error physics

28 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density 1D density distribution δ()

29 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density 1D density distribution δ()

30 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density 1D density distribution δ() AMR level #4 AMR level #3 AMR level #2 AMR level #1 domain grid

31 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density 1D density distribution δ() AMR level #4 AMR level #3 AMR level #2 AMR level #1 domain grid

32 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density: - refine regions of high density truncation error: physics:

33 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density: - refine regions of high density truncation error: - refine regions of large truncation errors i R k,l,m i = ΔΦ k,l,m ρ k,l,m ετ k,l,m with i [ ] ΔΦ k,l,m i Τ k,l,m = P Δ( RΦ k,l,m ) ( ) physics:

34 Adaptive Mesh Refinement adaptive mesh refinement refinement criterion density: - refine regions of high density truncation error: - refine regions of large truncation errors i R k,l,m i = ΔΦ k,l,m ρ k,l,m ετ k,l,m with i [ ] ΔΦ k,l,m i Τ k,l,m = P Δ( RΦ k,l,m ) ( ) physics: - compare grid spacing against local critical wavelength π Δ < ε λ with λ = c s Gρ

35 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

36 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes gravity generating refinements density assignment solving Poisson s equation handling irregular grids adaptive leap-frog integration

37 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes gravity generating refinements density assignment solving Poisson s equation handling irregular grids adaptive leap-frog integration

38 AMR for N-body gravity tends to clump matter together...

39 AMR for N-body gravity tends to clump matter together...

40 AMR for N-body gravity tends to clump matter together...

41 AMR for N-body gravity tends to clump matter together... introduce finer grids where needed and gain a factor of 2 in accuracy (in regions of interest)

42 AMR for N-body gravity tends to clump matter together... introduce finer grids where needed and gain a factor of 2 in accuracy (in regions of interest) factor 2 not mandatory, but most common choice

43 AMR for N-body gravity tends to clump matter together... introduce finer grids where needed periodic boundary conditions isolated boundary conditions

44 AMR for N-body gravity tends to clump matter together... introduce finer grids where needed...and update the grids where and when necessary

45 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes gravity generating refinements density assignment solving Poisson s equation handling irregular grids adaptive leap-frog integration

46 AMR Codes generating refinements N-body simulations: number of particles per cell refinement criterion: 6 particles/cell AMR for N-body

47 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell loop through given grid generating refinement by checking each individual cell N-body simulations: AMR for N-body

48 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell N-body simulations: AMR for N-body

49 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell N-body simulations: AMR for N-body

50 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell N-body simulations: AMR for N-body

51 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell N-body simulations: AMR for N-body

52 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell N-body simulations: AMR for N-body

53 AMR Codes generating refinements number of particles per cell refinement criterion: 6 particles/cell stop when no more cell require splitting N-body simulations: AMR for N-body

54 AMR for N-body generating refinements N-body simulations: number of particles per cell Note: in this scheme we split the volume of a coarse cell into eight equal sub-cells => non-cospatial scheme refinement criterion: 6 particles/cell

55 AMR for N-body generating refinements interpolation between grids: f ( i ) = F( i ) + ʹ F ( i )Δ F = value on coarse grid f = value on fine grid

56 AMR for N-body generating refinements interpolation between grids: f ( i ) = F( i ) + ʹ F ( i )Δ co-spatial vs. non-cospatial F( i ) F( i ) f ( i ) = F( i ) f ( i+1/ 2 ) = F( i ) + F ʹ ( i ) Δ 2 f ( i 1/ 2 ) = F( i ) F ʹ ( i ) Δ 2 f ( i+1/ 2 ) = F( i ) + F ʹ ( i ) Δ 2

57 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes gravity generating refinements density assignment solving Poisson s equation handling irregular grids adaptive leap-frog integration

58 AMR for N-body density assignment (co-spatial scheme)

59 AMR for N-body density assignment (co-spatial scheme)

60 AMR for N-body density assignment (co-spatial scheme) TSC mass assignment

61 AMR for N-body density assignment (co-spatial scheme) unproblematic:

62 AMR for N-body density assignment (co-spatial scheme) unproblematic:

63 AMR for N-body density assignment (co-spatial scheme) problematic:

64 AMR for N-body density assignment (co-spatial scheme) problematic:

65 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

66 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

67 AMR for N-body density assignment (co-spatial scheme) transfer particles to refinement

68 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

69 AMR for N-body density assignment (co-spatial scheme) density on coarse grid assign density on coarse grid

70 AMR for N-body density assignment (co-spatial scheme) density on coarse grid nodes with zero density

71 AMR for N-body density assignment (co-spatial scheme) density on coarse grid nodes carrying the density contribution from particles outside refinement => to be transferred to refinement nodes

72 AMR for N-body density assignment (co-spatial scheme) density on coarse grid remember nodes carrying the density contribution from particles outside refinement => to be transferred to refinement nodes

73 AMR for N-body density assignment (co-spatial scheme) density on coarse grid same nodes still missing density from particles on refinement edge => to be derived from refinement density

74 AMR for N-body density assignment (co-spatial scheme) density on coarse grid remember nodes still missing density from particles on refinement edge

75 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

76 AMR for N-body density assignment (co-spatial scheme) density on refinement grid assign density on refinement grid

77 AMR for N-body density assignment (co-spatial scheme) density on refinement grid refinement nodes still missing the density contribution from particles outside refinement

78 AMR for N-body density assignment (co-spatial scheme) density on refinement grid remember refinement nodes still missing the density contribution from particles outside refinement

79 AMR for N-body density assignment (co-spatial scheme) density on refinement grid all refinement nodes carry information required by coarse nodes

80 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

81 AMR for N-body density assignment (co-spatial scheme) density on coarse grid nodes carrying the density contribution from particles outside refinement

82 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

83 AMR for N-body density assignment (co-spatial scheme) density on coarse grid all refinement nodes carry information required by coarse nodes

84 AMR for N-body density assignment (co-spatial scheme) steps required to get density correct on both coarse and fine grid 1. transfer particles from coarse to fine grid 2. assign coarse particles to coarse grid 3. assign fine particles to refinement grid 4. temporarily store borderline density 5. inject refinement density to coarse grid 6. add borderline density to refinement

85 AMR for N-body density assignment (co-spatial scheme) density on refinement grid refinement nodes still missing the density contribution from particles outside refinement

86 AMR for N-body density assignment (co-spatial scheme) density finally correct on both levels

87 AMR for N-body density assignment (co-spatial scheme) density finally correct on both levels phew...

88 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes gravity generating refinements density assignment solving Poisson s equation handling irregular grids adaptive leap-frog integration

89 AMR for N-body solving Poisson s equation 16 3 boundary values 32 3 boundary values 64 3 boundary values 128 3

90 AMR for N-body adaptive mesh refinement cover simulation with regular domain grid create AMR hierarchy: generate fine grid by comparing each node against some refinement criterion " recursive procedure assign density on all grids solve Poisson s equation on regular domain grid (FFT is fastest ) loop over all refinement levels: interpolate potential down from parent level rela potential until converged (keeping boundary values fied) " this will give the correct potential on all (refinement) grids

91 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

92 AMR Codes Irregular Grids handling refinements refinement grids can have a highly irregular shape and consist of multiple spatially disjunctive blocks

93 AMR Codes Irregular Grids handling refinements how to manage and maneuver such irregular grids? refinement grids can have a highly irregular shape and consist of multiple spatially disjunctive blocks

94 Irregular Grids handling regular grids (1D) = struct { N = 16 node[0].rho, =3 node[1].rho, =6, node[n-1].rho, =48 float rho; } node;

95 Irregular Grids handling regular grids (1D) = struct { N = 16 float rho; } node; node[0].rho, =3 node[1].rho, =6, node[n-1].rho, =48 unique mapping between array inde i and spatial position possible

96 Irregular Grids handling irregular grids (1D) = N = 9 struct { float rho; } node;

97 Irregular Grids handling irregular grids (1D) = N = 9 node[0]. = 12 node[1]. = 15 node[2]. = 18 node[3]. = 21 node[4]. = 24 node[5]. = 27 node[6]. = 36 node[7]. = 39 node[8]. = 42 struct { long ; float rho; } node;

98 Irregular Grids handling irregular grids (1D) = N = 9 node[0]. = 12 node[1]. = 15 node[2]. = 18 node[3]. = 21 node[4]. = 24 node[5]. = 27 node[6]. = 36 node[7]. = 39 node[8]. = 42 struct { long ; float rho; } node; no unique mapping between array inde i and spatial position possible

99 Irregular Grids handling irregular grids (1D) = N = 9 node[0]. = 12 node[1]. = 15 node[2]. = 18 node[3]. = 21 node[4]. = 24 node[5]. = 27 node[6]. = 36 node[7]. = 39 node[8]. = 42 struct { long ; float rho; } node; no unique mapping between array inde i and spatial position possible generate a meta-structure storing the geometry of the grid

100 Irregular Grids handling irregular grids (1D) quad s = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { float rho; } node;

101 Irregular Grids handling irregular grids (1D) quad s the memory for each array of nodes needs to be consecutively malloc() ed = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { float rho; } node;

102 Irregular Grids handling irregular grids (1D) quad s the memory for each array of nodes needs to be consecutively malloc() ed = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { float rho; } node;

103 Irregular Grids handling irregular grids (1D) quad s the memory for each array of nodes needs to be consecutively malloc() ed = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { this node is addressed via (QUAD.node)+0 float rho; } node;

104 Irregular Grids handling irregular grids (1D) quad s the memory for each array of nodes needs to be consecutively malloc() ed = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { this node is addressed via (QUAD.node)+1 float rho; } node;

105 Irregular Grids handling irregular grids (1D) quad s the memory for each array of nodes needs to be consecutively malloc() ed = QUAD = (*node, 12, 6, *net) QUAD = (*node, 36, 3, *net) NULL struct { this node is addressed via (QUAD.node)+2 float rho; } node;

106 Irregular Grids handling irregular grids (2D) quad s

107 Irregular Grids handling irregular grids (2D) quad s the memory for each quad array needs to be malloc() ed consecutively

108 Irregular Grids handling irregular grids (2D) quad s store grid structures as a consecutive memory block each grid points to the first yquad which in turns gives access to all nodes grid #1 grid #1 grid #2 grid #3

109 Irregular Grids handling irregular grids (2D) quad s store grid structures as a consecutive memory block each grid points to the first yquad which in turns gives access to all nodes grid #2 grid #1 grid #2 grid #3 (pointer, y=1, length=3, net) (pointer, y=1, length=3, NULL) +1 (pointer, =1, length=2, net) (pointer, =1, length=2, NULL) (pointer, =4, length=2, net) (pointer, =1, length=2, net) (pointer, =5, length=2, NULL) (pointer, =4, length=2, net) (pointer, =9, length=1, NULL) (pointer, =8, length=2, NULL) (pointer, =7, length=3, NULL)

110 Irregular Grids handling irregular grids (3D) quad s too complicated to sketch

111 Irregular Grids handling irregular grids (3D) quad s C-code eample of how to loop over all nodes attached to a grid for (zquad=grid.first_zquad; zquad = NULL; zquad=zquad->net) { for (yquad=zquad->first_yquad; yquad < yquad->pointer+yquad->length; yquad++) for (iyquad=yquad; iyquad = NULL; iyquad=iyquad->net) { for (quad=yquad->first_quad; quad < quad->pointer+quad->length; quad++) for (iquad=quad; iquad = NULL; iquad=iquad->net) { for (node=iquad->pointer; node < iquad->+iquad->length; node++) { /* the node is at your disposal */ density = node->density; potential = node->potential; forcex = node->force[x]; for(part=node->first_particle; part = NULL; part=part->net) /* use particle structure to access particle position, velocity, etc. */ }}} loc = location of first quad

112 Irregular Grids handling irregular grids quad s drawback: no direct access to neighbouring nodes?

113 Irregular Grids handling irregular grids FTT other schemes possible: struct { NODE *daughter; float rho; } node; Fully-Threaded-Tree (FTT) by Khokhlov, 1998, J. Comp. Phy. 143, 519 (used with Andrey Kravtsov s ART code )

114 Irregular Grids handling irregular grids FTT other schemes possible: struct { NODE *daughter; float rho; } node; each cell stores pointers to 1 st daughter daughters are malloc() ed consecutively Fully-Threaded-Tree (FTT) by Khokhlov, 1998, J. Comp. Phy. 143, 519 (used with Andrey Kravtsov s ART code )

115 Itinerary mesh refinements adaptive mesh refinement adaptive mesh refinement for N-body codes handling irregular grids adaptive leap-frog integration

116 Adaptive Leap-Frog Integration full set of equations collisionless matter (e.g. dark matter) d DM = v DM dt leap-frog integration d v DM = φ dt AMR solver Poisson s equation Δφ = 4πGρ tot collisional matter (e.g. gas) ideal gas equations ρ t + ρ v ( ) = 0 p = ( γ 1)ρε ( ρ v ) t + ρ v v + p + 1 2µ B2 " 1 1 B B µ = ρ ( φ) ρε = ρe 1 2 ρv 2 ( ρe) t + ρe + p + 1 2µ B2 v 1 µ v B [ ] B = ρ v ( φ) + ( Γ L) Mawell s equation B t = v B ( )

117 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy AMR grids particles Δφ = 4πGρ tot d DM dt d v DM dt = v DM = φ

118 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy move particles on fine grids with smaller time step to better resolve the dynamics, too AMR grids particles Δφ = 4πGρ tot d DM dt d v DM dt = v DM = φ

119 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy fully recursive approach: fine grid coarse grid

120 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy fully recursive approach: Drift-Kick-Drift variant of the leap-frog integrator: time synchronisation between different grid levels rather than leap-frogging

121 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy fully recursive approach: 1. fine grid step: Drift : Kick : Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt p n +1/ 2 = p n Φ n +1/ 4 n +1/ 2 = n +1/ 4 + t n +Δt / 2 t n dt t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4

122 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy fully recursive approach: 2. coarse grid step: Drift : Kick : Drift : n +1/ 2 = n + t n +Δt / 2 p n p n +1 = p n Φ n +1/ 2 t n n +1 = n +1/ 2 + p n +1 dt t n +Δt t n t n +Δt dt dt t n +Δt / 2

123 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy fully recursive approach: 3. fine grid step: Drift : Kick : Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 p n +1 = p n +1/ 2 Φ n +3 / 4 n +1 = n +3 / 4 + p n +1 t n +Δt t n +Δt dt t n +Δt / 2 dt t n +3Δt / 4

124 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy

125 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 1. fine grid DKD step: Drift : Kick : Drift : n +1/ 4 = n + t n +Δt / 4 p n t n dt p n +1/ 2 = p n Φ n +1/ 4 n +1/ 2 = n +1/ 4 + t n +Δt / 2 t n dt t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4

126 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 1. fine grid DKD step: Drift : Kick : Drift : n +1/ 4 = n + t n +Δt / 4 p n t n dt p n +1/ 2 = p n Φ n +1/ 4 n +1/ 2 = n +1/ 4 + t n +Δt / 2 t n dt t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4

127 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 2. coarse grid DKD step: Drift : Kick : Drift : n +1/ 2 = n + t n +Δt / 2 p n t n p n +1 = p n Φ n +1/ 2 n +1 = n +1/ 2 + p n +1 dt t n +Δt t n t n +Δt dt dt t n +Δt / 2

128 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 2. coarse grid DKD step: Drift : Kick : Drift : n +1/ 2 = n + t n +Δt / 2 p n t n p n +1 = p n Φ n +1/ 2 n +1 = n +1/ 2 + p n +1 dt t n +Δt t n t n +Δt dt dt t n +Δt / 2

129 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 3. fine grid DKD step: Drift : Kick : Drift : n +3 / 4 = n +1/ 2 + t n +3Δt / 4 p n dt t n +Δt / 2 p n +1 = p n +1/ 2 Φ n +3 / 4 n +1 = n +3 / 4 + p n +1 t n +Δt t n +Δt dt t n +Δt / 2 dt t n +3Δt / 4

130 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 3. fine grid DKD step: Drift : Kick : Drift : n +3 / 4 = n +1/ 2 + t n +3Δt / 4 p n dt t n +Δt / 2 p n +1 = p n +1/ 2 Φ n +3 / 4 n +1 = n +3 / 4 + p n +1 t n +Δt t n +Δt dt t n +Δt / 2 dt t n +3Δt / 4

131 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy 3. fine grid DKD step: Drift : Kick : Drift : n +3 / 4 = n +1/ 2 + t n +3Δt / 4 p n dt t n +Δt / 2 p n +1 = p n +1/ 2 Φ n +3 / 4 n +1 = n +3 / 4 + p n +1 t n +Δt t n +Δt dt t n +Δt / 2 dt t n +3Δt / 4 what about particles crossing grid boundaries?

132 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4

133 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries particles can cross boundary during any of these steps Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4

134 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

135 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

136 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

137 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

138 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

139 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 2. coarse-grid drift 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 un-drift and move with coarse grid time step to t n+1

140 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 2. coarse-grid drift do nothing as particle has reached t n+1 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4

141 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 3. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 keep on drift ing as it will bring the particle to t n+1

142 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 3. fine-grid step 1. Drift : n +1/ 4 = n + p n t n +Δt / 4 t n dt 3. Drift : n +3 / 4 = n +1/ 2 + p n t n +3Δt / 4 dt t n +Δt / 2 2. Drift : n +1/ 2 = n +1/ 4 + t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 4. Drift : n +1 = n +3 / 4 + p n +1 t n +Δt dt t n +3Δt / 4 keep on drift ing as it will bring the particle to t n+1

143 Adaptive Leap-Frog Integration moving particles on the AMR hierarchy particles crossing grid boundaries 1. Drift : 2. Drift : n +1/ 4 = n + p n n +1/ 2 = n +1/ 4 + t n +Δt / 4 t n dt t n +Δt / 2 p n +1/ 2 dt t n +Δt / 4 3. Drift : 4. Drift : n +3 / 4 = n +1/ 2 + p n n +1 = n +3 / 4 + p n +1 t n +3Δt / 4 all particles have now moved from t n to t n+1 and the refinements will be re-created... dt t n +Δt / 2 t n +Δt dt t n +3Δt / 4

144 Pseudo C-Code Step(dt, CurrentGrid) { NewGrid = Refine(CurrentGrid); if(newgrid) { Step(dt/2, NewGrid); } MoveParticles(dt, CurrentGrid); } if(newgrid) { Step(dt/2, NewGrid); DestroyGrid(NewGrid);}

145 AMIGA