Modeling Myths. Modeling Facts

Transcription

1 Spiegelman: Columbia University, February 8, 22 Modeling Myths Modeling is Difficult Numerical models can do everything Numerical models will solve your problems Modeling Facts Modeling is simple book-keeping (plus a bit of magic) There are no black boxes! Numerical models have a life of their own and make their own problems

2 Spiegelman: Columbia University, February 8, 22 2 Used carefully, numerical solutions are a very powerful tool for gaining insight into possible physical processes. Used indiscriminately, they become an intellectual black-hole. We need to emphasize the two end-member approaches to modeling. The Kitchen Sink Approach: Throw everything into it and hope something useful comes out (BAD IDEA) 2. The Model Problem Approach: Gain insight by developing simple model problems that balance interesting behaviour with comprehensibility (GOOD IDEA but takes finesse).

3 Spiegelman: Columbia University, February 8, 22 3 This course will stress the 2nd approach and will emphasize the following axioms Be problem driven Understand your problem Keep it simple Never model more than you can understand (or observe?) Avoid the reality trap (models reality ) Choose your techniques to mimic the underlying physics. The only successful model is an insightful model.

4 Spiegelman: Columbia University, February 8, 22 4 Or more succinctly Think before you model Think while you model Think after you model

5 Spiegelman: Columbia University, February 8, 22 5 Some Earth science problems you ought to know (maybe). Thermal convection (2-D Rayleigh-Benard) 2. The Lorenz Equations ( Chaos and ODE s) 3. The shallow water equations 4. Seismic wave propagation 5. Flow in Porous Media

6 Spiegelman: Columbia University, February 8, 22 6 Thermal Convection (2-D Rayleigh Benard Convection) Pr T t + V T = 2 T ] [ ω t + V ω 2 ψ = ω = 2 ω Ra T x V = ψk, ω =( V) k Ra = α Tgd3 νκ, Pr = ν κ

7 Spiegelman: Columbia University, February 8, 22 7 The Lorenz Equations and chaos Let ψ(t, x) =W (t) sin(aπx) sin(πz) T (t, x) =( z)+t (t) cos(aπx) sin(πz)+t 2 (t) sin(2πz) Then RB convection becomes dw dt dt dt dt 2 dt = Pr(T W ) = WT 2 + rw T = WT bt 2

8 Spiegelman: Columbia University, February 8, Lorenz Equations Ra=28 (W,T,T 2 )=(,,) 4. W(t) T (t) T 2 (t) Variables Time

9 Spiegelman: Columbia University, February 8, 22 9 Linearized Shallow water equations equatorial β plane v t + βyk v = g η η t + (Hv) = With forcing and dissipation (ala Cane-Zebiak) v t + βyk v = gh η + τ /ρ rv η t + (Hv) = rη

10 Spiegelman: Columbia University, February 8, 22 Sea Surface temperature anomalies LDEO2 El Niño model Sea Surface temperature anomalies Latitude 3 S 3 N 5 E 8 5 W 2 W 9 W Longitude Sep C -.6 C -.4 C -.2 C C.2 C.4 C.6 C.8 C C.2 C.4 C temperature anomaly Latitude 3 S 3 N 5 E 8 5 W 2 W 9 W Longitude Dec98 - C - C C C C.5 C temperature anomaly

11 Spiegelman: Columbia University, February 8, 22 Seismic Wave propagation σ ij t ρ V t = σ + f ( Vi = µ + V ) j + λ Vδ ij x j x i Solutions by Pseudo-spectral techniques ala Gustavo Correa (LDEO)

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14 Spiegelman: Columbia University, February 8, 22 4 Flow in heterogeneous porous media K µ [ P ρ f g] = elastic rigid (rigid) -2-2 (elastic) log Permeability

15 Spiegelman: Columbia University, February 8, 22 5 Flow in deformable porous media (magma migration) (ρ f φ) t [ρ s ( φ)] t + (ρ f φv) =Γ + [ρ s ( φ)v] = Γ φ(v V) = k φ µ [ P ρ f g] P = [η V]+ [(ζ +4η/3) V]+G ρg k φ a2 φ n b

16 Spiegelman: Columbia University, February 8, 22 6 Flow in deformable porous media (2-D potential form, non-dimensional) φ t + V φ =( φ φ)c +Γ k φ C + C = k φ [ ωj ( φ φ)k]+γ δρ ρ f 2 U s = C 2 ω = g ρ η x 2 ψ s = ω

17 Spiegelman: Columbia University, February 8, 22 7 Non-linear Porosity waves Reactive Flow Localization time 4 height z/d 3 2 t= distance Earth Science Applications Thermal Structure β=7., U = 8. cm/yr, slab age=5. Ma width x/d 4 6 Fluid Flow depth (km) depth (km) mid-ocean ridges distance (km) subduction zones distance (km)

18 Spiegelman: Columbia University, February 8, 22 8 Direction fields and solutions for dc dt = c.5 Concentration Time

19 Spiegelman: Columbia University, February 8, 22 9 Simple Stepping Schemes. Concentration h Euler Step (first order step) True Solution Time Concentration 2 Mid-point Step (second order step) h 3 (euler) True Solution....5 Time

20 Spiegelman: Columbia University, February 8, th Order Runge-Kutta Step. Concentration 3 2 (mid-point) h 4 (euler) Runge-Kutta step (4th order) True Solution....5 Time

21 Spiegelman: Columbia University, February 8, 22 2 Bulirsch-Stoer Stepping using Richardson Extrapolation

22 Spiegelman: Columbia University, February 8, Transport by Characteristics: Particle based methods 2.5 t= t= concentration distance V(x) =.2x

23 Spiegelman: Columbia University, February 8, FTCS Explodes! 2. t= 2 3 t=4 concentration distance

24 Spiegelman: Columbia University, February 8, Staggered Leapfrog works (α <) α =.9 2. t= 2 3 t=4 concentration distance

25 Spiegelman: Columbia University, February 8, α =. 2 t= 2 t=2.53 concentration distance

26 Spiegelman: Columbia University, February 8, Staggered Leapfrog disperses (α =) t= distance t= concentration

27 Spiegelman: Columbia University, February 8, t= distance 2..5 t= concentration

28 Spiegelman: Columbia University, February 8, Simple Upwind diffuses (badly) (α =) 3 t= 2 concentration 2 t= distance

29 Spiegelman: Columbia University, February 8, Better schemes t= concentration t= concentration t= a distance b distance 3 3 concentration t= concentration t= (all identical) c distance d distance

30 Spiegelman: Columbia University, February 8, 22 3 Semi-Lagrangian Schemes: a recipe n+ true characteristic u(n+,j) n+/2 n t c(n) xx x u(n+/2) j x

31 Spiegelman: Columbia University, February 8, 22 3 Behaviour with non-constant velocity concentration 2 t= 2 3.9s staggered leapfrog.32s mpdata (ncor=3, i3rd=) concentration 2 2 t=.2s semi lagrangian 5.85s 2 3 distance pseudo spectral 2 3 distance

32 Spiegelman: Columbia University, February 8, Comparison of Pseudo-spectral and Semi-Lagrangian schemes 3 semi lagrangian (24 pts, α=2, t=.5s) 2 concentration pseudo spectral (256 pts,α=,t=4.98s) 2 3 distance

33 Spiegelman: Columbia University, February 8, Advection-diffusion: FTCS relative error. -.5 t= distance 3. t= 2.5 True solution Calculated solution Temperature distance

34 Spiegelman: Columbia University, February 8, Advection-diffusion: Crank-Nicholson.4.3 relative error t= distance 3. t= 2.5 True solution Calculated solution Temperature distance

35 Spiegelman: Columbia University, February 8, Advection-diffusion: Operator-Splitting MPDATA + CN (no corrections... upwind scheme). relative error. -. t= distance 3. t= 2.5 True solution Calculated Solution Temperature distance

36 Spiegelman: Columbia University, February 8, Advection-diffusion: Operator-Splitting MPDATA + CN.4 (one correction).2 relative error t= distance 3. t= 2.5 True solution Calculated Solution Temperature distance

37 Spiegelman: Columbia University, February 8, Advection-diffusion: Operator-Splitting MPDATA + CN Da Woiks! (ncor=3, i3rd=).2. relative error. t= distance 3. t= 2.5 True solution Calculated Solution Temperature distance

38 Spiegelman: Columbia University, February 8, Advection-diffusion: Operator-Splitting Semi-Lagrangian + CN α =2.5.2 relative error.. t= distance 3. t= 2.5 True solution Calculated Solution Temperature distance

39 Spiegelman: Columbia University, February 8, Advection-diffusion: All-in-one Semi-LagrangianCN α =2.5.2 relative error.. t= distance 3. t= 2.5 True solution Calculated Solution Temperature distance

40 Spiegelman: Columbia University, February 8, D control volume i,j+ F y (i,j+/2) F x (i+/2,j) i-,j i,j i+,j i,j-

41 Spiegelman: Columbia University, February 8, 22 4 Boundary condition pointers iout(2,2) side 2 iout(,) side side 2 iout(2,) dir 2 (j) dir (i) iout(,2) side

42 Spiegelman: Columbia University, February 8, Bicubic interpolation j+2 x j+ x + (ri,rj) j (i,j) x j- x i- i i+ i+2

43 Spiegelman: Columbia University, February 8, Some useful 2-D advection fields Rigid body rotation vs. a shear cell.9 a b

44 Spiegelman: Columbia University, February 8, Rigid Body rotation test: a methods sampler

45 Spiegelman: Columbia University, February 8, a.... b.... c.... d.... e... f...

46 Spiegelman: Columbia University, February 8, Rigid Body rotation test: a methods sampler

47 Spiegelman: Columbia University, February 8, stag-leap upwind mpdata mpdata mpdata stag-leap

48 .5 Spiegelman: Columbia University, February 8, Shear cell test Staggered Leapfrog vs. Upwind a b......

49 .5 Spiegelman: Columbia University, February 8, Shear cell test Mpdata (with everything) vs. High res Staggered Leapfrog...5 c d......

50 Spiegelman: Columbia University, February 8, 22 5 The winner! Semi-Lagrangian behaviour

51 Spiegelman: Columbia University, February 8, a..... b....5 c d......

52 Spiegelman: Columbia University, February 8, D Diffusion (booooring...) 2-D FTCS scheme

53 5e-5 5e-5 5e e-5 5e-5.5 5e-5.5 5e-5 5e-5 5e-5 5e-5 Spiegelman: Columbia University, February 8, D Diffusion Errors for 2-D FTCS scheme e-5-5e-5-5e e e-5.5 5e-5-5e e-5 5e-5-5e-5-5e e e-5.5 5e e-5.5-5e e e e e e-5.2-5e-5.2-5e e e-5.2-5e-5-5e-5-5e-5-5e-5

54 Spiegelman: Columbia University, February 8, D Diffusion Errors for ADI scheme

55 Spiegelman: Columbia University, February 8, Structure of the 5-point Laplacian Operator as a sparse matrix nz = 65

56 Spiegelman: Columbia University, February 8, nz = 27

57 Spiegelman: Columbia University, February 8, Structure of the inverse of the 5-point Laplacian Operator as a sparse matrix

58 Spiegelman: Columbia University, February 8, Testing Laplace Solvers: the sin-cell test. Solution Errors

59 Spiegelman: Columbia University, February 8,

60 Spiegelman: Columbia University, February 8, 22 6 Convergence Behaviour of optimal SOR 4 2 L2 Norm of residual iterations

61 Spiegelman: Columbia University, February 8, 22 6 A nested Multi-level Grid (A 3-level grid)

62 Spiegelman: Columbia University, February 8, V-cycles and Full Multi-Grid (FMG) cycles V-cycle 4 S 3 R R coarsest grid 2 R R R R finest grid S S S S 4 coarsest grid R R R R R R 3 R R R R 2 FMG-Vcycle R R finest grid

63 Spiegelman: Columbia University, February 8, Multi-Grid storage scheme ala Briggs A u rhs res 5 grid ip() ip(2) ip(3) ip(4)

64 Spiegelman: Columbia University, February 8, A V-cycle in the Briggs Scheme Going up!

65 Spiegelman: Columbia University, February 8, grid u rhs res then calculate residual initial Guess relax Npre times then restrict ip() ip(2) ip(3) ip(4) fine u rhs res relax Npre times then calculate residual then restrict u rhs res u solve

66 Spiegelman: Columbia University, February 8, A V-cycle in the Briggs Scheme Coming Down!

67 Spiegelman: Columbia University, February 8, grid u rhs interp relax Npost times ip() ip(2) ip(3) ip(4) add back interpolate correction coarse u rhs interp relax add interpolate u rhs interp much improved guess relax add interpolate fine

68 Spiegelman: Columbia University, February 8, The Big test! Timing and errors for solving Poisson problem with Dirichlet Boundaries on 3 2 a 2-2 sin cell test ce s, O3 q o qa c qau odb db Fishpak Y2M MG Vcycle FMG SOR time (cpu seconds) Ni= Total Grid points

69 Spiegelman: Columbia University, February 8, ce s, O3 q o qa c qau odb db average error Fishpak Y2M MG Vcycle FMG SOR Total Grid points

70 Spiegelman: Columbia University, February 8, 22 7 General Conservation of Mass for chemistry t ρ s( φ)c s + [ρ s ( φ)vc s ]= I + D s () t ρ f φc f + [ρ f φvc f ] = I + D f (2) For Fractional melting I =(c s /D)Γ, therefore t ρ s( φ)c s + [ρ s ( φ)vc s ]= cs D Γ (3) t ρ f φc f + [ρ f φvc f ] = cs D Γ (4)

71 Spiegelman: Columbia University, February 8, 22 7 Expanding By chain rule gives But Conservation of total mass is [ ] c s t ρ s( φ)+ [ρ s ( φ)v] + [ ] c s ρ s ( φ) t + V cs = cs D Γ [ ] c f t ρ f φ + [ρ f φv] + [ c f ρ f φ t + v cf ] = cs D Γ t ρ s( φ)+ [ρ s ( φ)v] = Γ (5) t ρ f φ + [ρ f φv] =Γ (6)

72 Spiegelman: Columbia University, February 8, Therefore we can write the problem as c s Γ+ρ s ( φ) D sc s Dt = cs D Γ (7) c f Γ+ρ f φ D fc f Dt = cs D Γ (8)

73 Spiegelman: Columbia University, February 8, Rearranging and Scaling with Yields D s c s ( ) = c s Dt D Γ ρ s ( φ) D f c f ( ) c s Γ = Dt D cf ρ f φ φ = φ φ t = w δ t Γ = ρ sφ w δ c s = c s c s c f = cs D cf Γ (9) ()

74 Spiegelman: Columbia University, February 8, D s c s Dt φ = ( φ φ) cs D f c f Dt = ( c s c f ) ρ s Γ ρ f φ ( ) D Γ () (2)

75 Spiegelman: Columbia University, February 8, General Semi-Lagrangian solution of is or Dc Dt c + c t Numerics = g(c, x,t) = 2 [ g + + g ] c + = c + t [ g + + g ] 2 Therefore for solid concentration can discretize D s c s Dt = φ ( φ φ) cs ( ) D Γ as c s+ c s (s ) = (Ac s ) + (Ac s ) (s )

76 Spiegelman: Columbia University, February 8, where therefore A = tφ 2( φ φ) ( D )Γ

77 Spiegelman: Columbia University, February 8, Same for melt concentration; D f c f Dt = ( c s c f ) ρ s Γ ρ f φ (3) Becomes c f + c f (f ) =[B(c s c f )] + +[B(c s c f )] (f ) where Therefore B = tρ sγ 2ρ f φ ( + B + )c f + =( B (f ) )c f (f ) +(Bc s ) + +(Bc s ) (f ) or [( B (f ) )c f (f ) +(Bc s ) + +(Bc s ) (f ) ] c f + = Final Update Scheme: ( + B + )

78 Spiegelman: Columbia University, February 8, c s+ = ( A(s ) ) ( + A + ) cs (s ) ] [( B (f ) )c f (f ) +(Bc s ) + +(Bc s ) (f ) c f + = ( + B + )