Computational Astrophysics

Transcription

1 Computatonal Astrophyscs Solvng for Gravty Alexander Knebe, Unversdad Autonoma de Madrd

2 Computatonal Astrophyscs Solvng for Gravty the equatons full set of equatons collsonless matter (e.g. dark matter d x DM = v DM dt d v DM = φ dt Posson s equaton Δφ = 4πGρ tot collsonal matter (e.g. gas deal gas equatons ρ t + ρ v ( = 0 p = ( γ 1ρε ( ρ v t & + ρ v v & ( + ( p + 1 ' ' 2µ B2 " B B + * µ * = ρ ( φ ρε = ρe 1 2 ρv 2 ( ρe t,% +. ' ρe + p + 1 -& 2µ B2 ( * v 1 µ v B [ ] B / 1 0 = ρ v ( φ + ( Γ L Maxwell s equaton B t = v B (

3 Computatonal Astrophyscs Solvng for Gravty the equatons full set of equatons collsonless matter (e.g. dark matter d x DM = v DM dt d v DM = φ and the force dt Posson s equaton Δφ = 4πGρ tot collsonal matter (e.g. gas deal gas equatons ρ t + ρ v ( = 0 p = ( γ 1ρε ( ρ v t & + ρ v v & ( + ( p + 1 ' ' 2µ B2 " B B + * µ * = ρ ( φ ρε = ρe 1 2 ρv 2 ( ρe t,% +. ' ρe + p + 1 -& 2µ B2 ( * v 1 µ v B [ ] B / 1 0 = ρ v ( φ + ( Γ L Maxwell s equaton B t = v B (

4 Computatonal Astrophyscs Solvng for Gravty Posson s Equaton Posson s equaton F ( x = m Φ( x ΔΦ( x = 4πGρ( x

5 Computatonal Astrophyscs Solvng for Gravty Posson s Equaton Posson s equaton F ( x = m Φ( x ΔΦ( x = 4πGρ( x partcle approach F ( x Gm = m j (x x j ( x 3 x j j grd approach ΔΦ( x, j,k = 4πGρ( x, j,k F ( x, j,k = m Φ( x, j,k ( x, j,k =poston of centre of grd cell (, j,k

6 Computatonal Astrophyscs Solvng for Gravty Posson s equaton F ( x = m Φ( x ΔΦ( x = 4πGρ( x weapon of choce: tree codes grd approach ΔΦ( x, j,k = 4πGρ( x, j,k F ( x, j,k = m Φ( x, j,k Posson s Equaton partcle approach F ( x Gm = m j (x x j ( x 3 x j j ( x, j,k =poston of centre of grd cell (, j,k

7 Computatonal Astrophyscs Solvng for Gravty Posson s Equaton Posson s equaton F ( x = m Φ( x ΔΦ( x = 4πGρ( x partcle approach F ( x Gm = m j (x x j ( x 3 x j j grd approach ΔΦ( x, j,k = 4πGρ( x, j,k F ( x, j,k = m Φ( x, j,k ( x, j,k =poston of centre of grd cell (, j,k weapon of choce: AMR codes

8 Computatonal Astrophyscs Solvng for Gravty the partcle-mesh (PM method

9 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m

10 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m

11 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x N N

12 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x sounds lke a waste of tme and computer resources, but exceptonally fast n practce

13 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x

14 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m example: 1 partcle on 1 dmensonal grd M(g k = mw (d d = x g k ρ(g k = M(g k H partcle poston x grd pont g k H m x axs

15 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m example: 1 partcle on 1 dmensonal grd M(g k = mw (d ρ(g k = M(g k H d = x g k mass assgnment functon partcle poston x grd pont g k H m x axs

16 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m example: 1 partcle on 1 dmensonal grd herarchy of mass assgnment schemes: - Nearest-Grd-Pont NGP - Could-In-Cell CIC - Trangular-Shaped Cloud TSC - partcle poston x grd pont g k H m x axs

17 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Nearest-Grd-Pont (NGP: mass assgnment functon: # W (d = $ 1 d H /2 % 0 otherwse H m x axs

18 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Nearest-Grd-Pont (NGP: partcle shape: S(x = δ(x mass assgnment functon: # W (d = $ 1 d H /2 % 0 otherwse H m x axs

19 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Cloud-In-Cell (CIC: mass assgnment functon: $ 1 d & H W (d = % d H & 0 '& otherwse H m x axs

20 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Cloud-In-Cell (CIC: partcle shape: # S(x = $ 1 x H /2 % 0 otherwse mass assgnment functon: $ 1 d & H W (d = % d H & 0 '& otherwse H m x axs

21 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Trangular-Shaped-Cloud (TSC: * 3 4 d 2 # &, % ( d H mass assgnment functon:, $ H ' 2, W (d = 1# 3 % 2 2 d 2 + & H, ( $ H ' 2 d 3H 2,, -, 0 otherwse H m x axs

22 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m Trangular-Shaped-Cloud (TSC: partcle shape: $ & S(x = 1 x % x H H '& 0 otherwse * 3 4 d 2 # &, % ( d H mass assgnment functon:, $ H ' 2, W (d = 1# 3 % 2 2 d 2 + & H, ( $ H ' 2 d 3H 2,, -, 0 otherwse H m x axs

23 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m N partcles on 3 dmensonal grd d = x g k,l,m M( g N k,l,m = m W ( d x W ( d y W ( d z =1 ρ( g k,l,m = M( g k,l,m H 3

24 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes x ρ( g k,l,m N partcles on 3 dmensonal grd d = x g k,l,m M( g N k,l,m = m W ( d x W ( d y W ( d z =1 for every grd pont we need to loop over all N partcles ρ( g k,l,m = M( g k,l,m H 3

25 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes - n practce x ρ( g k,l,m rather loop over all partcles and assgn them to the approprate grd ponts, because the mappng x g k s rather easy

26 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes - n practce x ρ( g k,l,m example for CIC assgnment n 2D: x contrbutes ts mass m to the 4 closest grd ponts : m

27 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes - n practce x ρ( g k,l,m example for CIC assgnment n 3D: x contrbutes ts mass m to the 8 closest grd ponts : g k,l,m +1 g k +1,l,m +1 g k,l,m g k +1,l,m g k,l +1,m +1 g k +1,l +1,m +1 g k,l +1,m g k +1,l +1,m

28 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes whch scheme to choose? NGP = stepwse force CIC = contnuous pecewse lnear force (1 grd pont (8 grd ponts TSC = contnuous force and frst dervatve (27 grd ponts

29 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes whch scheme to choose? NGP = too crude CIC = common choce ncreased smoothng of densty feld TSC = pretty smooth smoothng the densty feld wll lead to a bas n the forces but at the same tme decrease the varance bas = ( F( x F true ( x ε α var = F 2 ( x F( x 2 N β

30 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method densty assgnment schemes whch scheme to choose? NGP = too crude CIC = common choce TSC = pretty smooth smoothng the densty feld wll lead to a bas n the forces but at the same tme decrease the varance bas = ( F( x F true ( x ε α var = F 2 ( x F( x 2 N β (nterplay between N and ε: Nε 3 =const.

31 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x

32 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton ΔΦ k,l,m = ρ k,l,m relaxaton technque: FTT technque: applcable and usable for any dfferental equaton only applcable and usable for lnear dfferental equaton

33 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton ΔΦ k,l,m = ρ k,l,m

34 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton ΔΦ k,l,m = ρ k,l,m ΔΦ k,l,m = Φ k,l,m ' * ' Φ *, k,l,m, x, x, =, Φ k,l,m, y, y, Φ, k,l,m, ( z + ( z + ' *, ' = 1 x, Φ Φ * k + 1,l,m k 1,l,m 2 2,, Φ H y, Φ, k,l + 1,m k,l 1,m 2 2 Φ k,l,m + 1 Φ, k,l,m 1, ( ( z + = 1 ' Φ k + 1 Φ,l,m k 1,l,m H ( x x Φ k,l + 1 2,m y Φ k,l 1 2,m y + Φ k,l,m z Φ k,l,m 1 2 z = 1 ( Φ H 2 k +1,l,m 2Φ k,l,m + Φ k 1,l,m + Φ k,l +1,m 2Φ k,l,m + Φ k,l 1,m + Φ k,l,m +1 2Φ k,l,m + Φ k,l,m 1 *, +

35 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton ΔΦ k,l,m = ρ k,l,m dscretzed Posson s equaton Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

36 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton ΔΦ k,l,m = ρ k,l,m teratve soluton: Φ k,l,m +1 Φ k,l,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

37 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton k,l+1,m k-1,l,m k,l,m k+1,l,m k,l-1,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

38 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton relaxaton technque obtan teratve solver by dscretzng dfferental equaton k-1,l,m k,l+1,m k,l,m k,l-1,m k+1,l,m applcable to grds of arbtrary geometry dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

39 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque how to sweep through the grd? k,l+1,m k-1,l,m k,l,m k+1,l,m k,l-1,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

40 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque how to sweep through the grd?? dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

41 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque how to sweep through the grd?? dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

42 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque Gauss-Sedel sweeps: k,l+1,m k-1,l,m k,l,m k+1,l,m k,l-1,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

43 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque Gauss-Sedel sweeps: loop over all black cells loop over all red cells one teraton of the potental Φ k,l,m +1 Φ k,l,m how many teratons are necessary? dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

44 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: ΔΦ k,l,m = ρ k,l,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

45 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: ΔΦ k,l,m? ρ k,l,m & 0 dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

46 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: ΔΦ k,l,m? ρ k,l,m & 0 densty as gven by mass assgnment scheme densty as gven by currently best guess for Φ dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

47 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: ΔΦ k,l,m? ρ k,l,m & 0 resdual: R = ΔΦ k,l,m ρ k,l,m = sutable norm dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

48 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: ΔΦ k,l,m? ρ k,l,m & 0 R = ΔΦ k,l,m ρ k,l,m ετ = sutable norm tolerance error estmate dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

49 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ - truncaton error: error due to dscreteness of grd dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

50 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ - truncaton error: error due to dscreteness of grd estmaton "" " compare soluton on actual grd to soluton on coarser grd dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

51 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ = ε T k,l,m - truncaton error: [ ] ΔΦ k,l,m Τ k,l,m = P Δ( RΦ k,l,m ( RΦ k,l,m Δ RΦ k,l,m =Φ j,n,p 1 ( = ρ j,n, p ( [ ] = ρ k,l,m P Δ RΦ k,l,m restrcton to coarser grd prolongaton to fner grd dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

52 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ = ε T k,l,m - truncaton error: [ ] ΔΦ k,l,m Τ k,l,m = P Δ( RΦ k,l,m ( RΦ k,l,m =Φ j,n,p 1 Δ( RΦ k,l,m = ρ j,n, p [ ( ] = ρ k,l,m P Δ RΦ k,l,m? = ΔΦ k,l,m dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

53 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ = ε T k,l,m R εt dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

54 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque stoppng crteron: R = ΔΦ k,l,m ρ k,l,m ετ = ε T k,l,m R faster convergence? εt dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

55 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque convergence: R = ΔΦ k,l,m ρ k,l,m - slow convergence: R +1 R large-scale errors n Φ cannot be relaxed suffcently fast on the actual grd => use coarser grds to speed up convergence dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

56 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton relaxaton technque convergence: R = ΔΦ k,l,m ρ k,l,m - slow convergence: R +1 R mult-grd relaxaton technques => beyond the scope of ths lecture though dscretzed Posson s equaton +1 Φ k,l,m = 1 6 (Φ k +1,l,m + Φ k 1,l,m + Φ k,l +1,m + Φ k,l 1,m + Φ k,l,m +1 + Φ k,l,m 1 ρ k,l,m H 2

57 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton ΔΦ k,l,m = ρ k,l,m Green s functon method: - solve dfferental equaton by Fourer transformaton - applcable and usable for lnear dfferental equatons

58 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton fast fourer transform method Green s functon method ΔΦ = ρ Φ( x = G( x x $ ρ( x $ d 3 $ x ; G( x = 1 4π x

59 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton fast fourer transform method Green s functon method ΔΦ = ρ Φ( x = G( x x $ ρ( x $ d 3 $ x ; G( x = 1 4π x Φ = ρ G FFT " convoluton becomes multplcaton ˆ Φ = ˆ ρ ˆ G (wth G ˆ = 1 for Posson s equaton k 2

60 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton fast fourer transform method Green s functon method ΔΦ = ρ Φ( x = G( x x $ ρ( x $ d 3 $ x ; G( x = 1 4π x Φ = ρ G FFT " convoluton becomes multplcaton ˆ Φ = ˆ ρ ˆ G (wth G ˆ = 1 for Posson s equaton k 2 Φ FFT -1 (FFT demands a regular grd though

61 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton fast fourer transform method dscretzed Green s functon G ˆ ( k = 1 k 2 " G ˆ ( g k,l,m = # sn 2 % $ k x 2 1 & # ( + sn k & 2 y # k % ( + sn 2 z & % ( ' $ 2 ' $ 2 ' G ˆ ( g 0,0,0 = 0, k x = 2π k L, k y = 2π l L, k y = 2π m L

62 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton accuracy of ether relaxaton or FFT method to solve Posson s equaton?

63 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton

64 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton

65 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton the force s automatcally softened... (cf. tree-code lecture

66 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton the force s automatcally softened...but what f need to resolve smaller scales?

67 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton ntroducton of fner grds n hgh densty regons

68 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton AMR calculaton Yahag & Yosh (2001

69 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton AMR calculaton detals later Yahag & Yosh (2001

70 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton and what are these wggles?

71 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton and what are these wggles?

72 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method numercally ntegrate Posson s equaton pure PM calculaton force ansotropy and what are these wggles?

73 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x

74 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method obtanng the forces F ( g k,l,m = m Φ( g k,l,m F x ( g k,l,m = m Φ( g k +1,l,m Φ( g k 1,l,m 2H F y ( g k,l,m = m Φ( g k,l +1,m Φ( g k,l 1,m F z ( g k,l,m = m Φ( 2H g k,l,m +1 Φ( g k,l,m 1 2H k-1,l,m k,l+1,m k,l,m k,l-1,m k+1,l,m H = (current grd spacng

75 Computatonal Astrophyscs Partcle-Mesh Method Solvng for Gravty numercally ntegrate Posson s equaton ΔΦ( g k,l,m = 4πGρ( g k,l,m F ( g k,l,m = m Φ( g k,l,m 1. calculate mass densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x

76 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method nterpolatng the forces F ( g k,l,m F ( r

77 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method nterpolatng the forces F ( g k,l,m F ( r use the nverse of the mass assgnment scheme to nsure momentum conservaton and mnmze force ansotropes F( r = k l m F( g k,l,m W ( r g k,l,m n practce the trple sum s only over 8 (CIC or 27 (TSC cells

78 Computatonal Astrophyscs Solvng for Gravty Partcle-Mesh Method nterpolatng the forces F ( g k,l,m F ( r use the nverse of the mass assgnment scheme to nsure momentum conservaton and mnmze force ansotropes F( r = k l m F( g k,l,m W ( r g k,l,m *check by calculatng the total (perodc force: N F tot = F( r = =... = N =1 =1 k l m N N =... =1 j=1 k,l,m k #, l #, m# F( g k,l,m W ( r g k,l,m m m j W ( r H 3 g k,l,m G( g k,l,m g k #, l #, m # W ( r j g # ant-symmetrc = 0 (because of nvarance under coordnate nverson k, l #, m # PM scheme: F x ( g k,l,m = m Φ( g k+1,l,m Φ( g k 1,l,m 2H Φ( g k,l,m = G( g k,l,m g k #, l #, m # ρ( g # k# l# m# ρ( g k,l,m = M( g k,l,m H 3 N M( g k,l,m = m W ( r g k,l,m =1 k, l #, m #

79 Computatonal Astrophyscs Summary Solvng for Gravty Partcle-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 densty on grd 2. solve Posson s equaton on grd 3. dfferentate potental to get forces 4. nterpolate forces back to partcles x ρ( g k,l,m Φ( g k,l,m F ( g k,l,m F ( g k,l,m F ( x anyone fances to wrte a PM code as the project?

80 Computatonal Astrophyscs Solvng for Gravty Alexander Knebe, Unversdad Autonoma de Madrd