Computational High Frequency Wave Propagation

Transcription

1 Computational High Frequency Wave Propagation Olof Runborg CSC, KTH Isaac Newton Institute Cambridge, February 2007 Olof Runborg (KTH) High-Frequency Waves INI, / 52

2 Outline 1 Introduction, background 2 Geometrical optics (GO) 3 Formulations of GO and numerical methods 1 Eikonal equations 2 Ray tracing 3 Wavefront methods 4 Kinetic formulation 5 Phase space methods 4 Other models Olof Runborg (KTH) High-Frequency Waves INI, / 52

3 Wave equation Cauchy problem for scalar wave equation u tt c(x) 2 u = 0, (t, x) R + R d, u(0, x) = A(x)e iωφ(x ), u t (0, x) = ωb(x)e iωφ(x ), where c(x) (variable) speed of propagation. Helmholtz equation (u = v exp(iωt)) v + n(x) 2 ω 2 v = f (x), +radiation condition, where n(x) := 1/c(x) index of refraction, ω angular frequency. Scattering problem Similar versions for elastic wave equation, Maxwell equations, Schrödinger equation,.... Olof Runborg (KTH) High-Frequency Waves INI, / 52

4 Computational challenge Simulation at high frequencies a major challenge! I.e., high relative to size of computational domain in time and space, ω 1 T, ω c = k 1 X. (When 0 t T and 0 x X.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

5 Computational challenge Simulation at high frequencies a major challenge! I.e., high relative to size of computational domain in time and space, ω 1 T, ω c = k 1 X. (When 0 t T and 0 x X.) High frequency short wave length highly oscillatory solutions many gridpoints. Olof Runborg (KTH) High-Frequency Waves INI, / 52

6 Computational challenge Simulation at high frequencies a major challenge! I.e., high relative to size of computational domain in time and space, ω 1 T, ω c = k 1 X. (When 0 t T and 0 x X.) High frequency short wave length highly oscillatory solutions many gridpoints. Direct numerical solution resolves wavelength: #gridpoints ω d at least. Olof Runborg (KTH) High-Frequency Waves INI, / 52

7 Computational challenge Simulation at high frequencies a major challenge! I.e., high relative to size of computational domain in time and space, ω 1 T, ω c = k 1 X. (When 0 t T and 0 x X.) High frequency short wave length highly oscillatory solutions many gridpoints. Direct numerical solution resolves wavelength: #gridpoints ω d at least. Often unrealistic approach for applications in e.g. optics, electromagnetics, geophysics, acoustics,... Olof Runborg (KTH) High-Frequency Waves INI, / 52

8 Computational challenge cont. Challenge met by Efficient direct numerical methods (cost increases with ω for fixed accuracy) High frequency approximations (accuracy increases with ω for fixed cost) Prescribed tolerance methods for certain problems, [Bruno, Chandler-Wilde,... ] (cost independent of ω for fixed accuracy) Olof Runborg (KTH) High-Frequency Waves INI, / 52

9 Geometrical optics Helmholtz equation u + ω 2 n(x) 2 u = 0. Write solution on the form u(x) = A(x, ω)e iωφ(x). Olof Runborg (KTH) High-Frequency Waves INI, / 52

10 Geometrical optics Helmholtz equation u + ω 2 n(x) 2 u = 0. Write solution on the form u(x) = A(x, ω)e iωφ(x). (a) Amplitude A(x) (b) Phase φ(x) Olof Runborg (KTH) High-Frequency Waves INI, / 52

11 Geometrical optics Amplitude A and phase φ vary on a much coarser scale than u. (And varies little with ω.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

12 Geometrical optics Amplitude A and phase φ vary on a much coarser scale than u. (And varies little with ω.) Geometrical optics approximation considers A and φ as ω. Olof Runborg (KTH) High-Frequency Waves INI, / 52

13 Geometrical optics Amplitude A and phase φ vary on a much coarser scale than u. (And varies little with ω.) Geometrical optics approximation considers A and φ as ω. Good accuracy for large ω. Computational cost ω-independent. u(x) = A(x)e iωφ(x) + O(1/ω). Olof Runborg (KTH) High-Frequency Waves INI, / 52

14 Geometrical optics Amplitude A and phase φ vary on a much coarser scale than u. (And varies little with ω.) Geometrical optics approximation considers A and φ as ω. Good accuracy for large ω. Computational cost ω-independent. u(x) = A(x)e iωφ(x) + O(1/ω). Waves propagate as rays, c.f. visible light. Not all wave effects captured correctly (diffraction, caustics). Olof Runborg (KTH) High-Frequency Waves INI, / 52

15 Geometrical optics Amplitude A and phase φ vary on a much coarser scale than u. (And varies little with ω.) Geometrical optics approximation considers A and φ as ω. Good accuracy for large ω. Computational cost ω-independent. u(x) = A(x)e iωφ(x) + O(1/ω). Waves propagate as rays, c.f. visible light. Not all wave effects captured correctly (diffraction, caustics). More generally, multiple crossing waves: u(x) = N A n (x)e iωφn(x) + O(1/ω). n=1 Several amplitude and phase functions. Olof Runborg (KTH) High-Frequency Waves INI, / 52

16 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). Olof Runborg (KTH) High-Frequency Waves INI, / 52

17 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Olof Runborg (KTH) High-Frequency Waves INI, / 52

18 Limitations of geometrical optics Diffracted waves Olof Runborg (KTH) High-Frequency Waves INI, / 52

19 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Olof Runborg (KTH) High-Frequency Waves INI, / 52

20 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Creeping rays Olof Runborg (KTH) High-Frequency Waves INI, / 52

21 Limitations of geometrical optics Creeping rays Olof Runborg (KTH) High-Frequency Waves INI, / 52

22 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Creeping rays Olof Runborg (KTH) High-Frequency Waves INI, / 52

23 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Creeping rays Disappear in GO limit, but is of size 1/ω α, 0 < α < 1. (Hence 1/ω.) GTD used in numerical simulations by e.g. adding diffracted waves at corners. Olof Runborg (KTH) High-Frequency Waves INI, / 52

24 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Creeping rays Disappear in GO limit, but is of size 1/ω α, 0 < α < 1. (Hence 1/ω.) GTD used in numerical simulations by e.g. adding diffracted waves at corners. Caustics is another problem: Amplitude A infinite, but should be A ω α, 0 < α < 1. Olof Runborg (KTH) High-Frequency Waves INI, / 52

25 Limitations of geometrical optics Caustics Concentration of rays. A when ω. Olof Runborg (KTH) High-Frequency Waves INI, / 52

26 Limitations of geometrical optics GO approximation breaks down when the wavelength variations in c(x). By the geometrical theory of diffraction (GTD) ([Keller, 59]) errors can be much larger than O(1/ω) because of boundaries: Diffracted waves Creeping rays Disappear in GO limit, but is of size 1/ω α, 0 < α < 1. (Hence 1/ω.) GTD used in numerical simulations by e.g. adding diffracted waves at corners. Caustics is another problem: Amplitude A infinite, but should be A ω α, 0 < α < 1. Olof Runborg (KTH) High-Frequency Waves INI, / 52

27 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

28 Numerical and modeling issues Goal: Find phase (traveltime) and amplitude in a domain, on a regular grid. Issues: Multiple arrivals, crossing rays, superposition multivaluedness. Eulerian/Lagrangian model. Fixed/moving grids. PDE/ODE model. Complexity. Olof Runborg (KTH) High-Frequency Waves INI, / 52

29 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

30 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Olof Runborg (KTH) High-Frequency Waves INI, / 52

31 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Enter this into Helmholtz equation u + ω 2 n 2 u = 0, ω 2 (n 2 φ 2 )u + iω(2 φ a 0 + φa 0 )e iωφ(x) + O(ω 0 ) = 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

32 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Enter this into Helmholtz equation u + ω 2 n 2 u = 0, ω 2 (n 2 φ 2 )u + iω(2 φ a 0 + φa 0 )e iωφ(x) + O(ω 0 ) = 0. Equate coefficients of powers of ω to zero: Olof Runborg (KTH) High-Frequency Waves INI, / 52

33 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Enter this into Helmholtz equation u + ω 2 n 2 u = 0, ω 2 (n 2 φ 2 )u + iω(2 φ a 0 + φa 0 )e iωφ(x) + O(ω 0 ) = 0. Equate coefficients of powers of ω to zero: For ω 2 we get the eikonal equation φ = n(x), Olof Runborg (KTH) High-Frequency Waves INI, / 52

34 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Enter this into Helmholtz equation u + ω 2 n 2 u = 0, ω 2 (n 2 φ 2 )u + iω(2 φ a 0 + φa 0 )e iωφ(x) + O(ω 0 ) = 0. Equate coefficients of powers of ω to zero: For ω 2 we get the eikonal equation φ = n(x), For ω 1 we get the transport equation 2 φ a 0 + a 0 φ = 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

35 Eikonal- and transport equations Derivation Make the WKB ansatz u(x) = e iωφ(x) k=0 a k (x)(iω) k. Enter this into Helmholtz equation u + ω 2 n 2 u = 0, ω 2 (n 2 φ 2 )u + iω(2 φ a 0 + φa 0 )e iωφ(x) + O(ω 0 ) = 0. Equate coefficients of powers of ω to zero: For ω 2 we get the eikonal equation φ = n(x), For ω 1 we get the transport equation 2 φ a 0 + a 0 φ = 0. Discard rest of terms in series (ω ). Let A = a 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

36 Eikonal equation Hamilton Jacobi type nonlinear PDE. Can be solved efficiently on fixed Eulerian grids. Olof Runborg (KTH) High-Frequency Waves INI, / 52

37 Eikonal equation Hamilton Jacobi type nonlinear PDE. Can be solved efficiently on fixed Eulerian grids. Stationary version. φ = n(x). Fast marching [Sethian,Tsitsiklis] or fast sweeping methods [Zhao, Tsai, et al]. Olof Runborg (KTH) High-Frequency Waves INI, / 52

38 Eikonal equation Hamilton Jacobi type nonlinear PDE. Can be solved efficiently on fixed Eulerian grids. Stationary version. φ = n(x). Fast marching [Sethian,Tsitsiklis] or fast sweeping methods [Zhao, Tsai, et al]. Time-dependent version. Wave equation plus ansatz u(t, x) A(t, x)e iωϕ(t,x) give ϕ t + n(x) 1 ϕ = 0. Upwind, high-resolution (ENO, WENO) finite difference methods [Osher, Shu, et al] Olof Runborg (KTH) High-Frequency Waves INI, / 52

39 Eikonal equation Hamilton Jacobi type nonlinear PDE. Can be solved efficiently on fixed Eulerian grids. Stationary version. φ = n(x). Fast marching [Sethian,Tsitsiklis] or fast sweeping methods [Zhao, Tsai, et al]. Time-dependent version. Wave equation plus ansatz u(t, x) A(t, x)e iωϕ(t,x) give ϕ t + n(x) 1 ϕ = 0. Upwind, high-resolution (ENO, WENO) finite difference methods [Osher, Shu, et al] (Note, if IC and BC match, ϕ = φ t.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

40 Eikonal equation Properties Eikonal equation: φ = n(x), Olof Runborg (KTH) High-Frequency Waves INI, / 52

41 Eikonal equation Properties Eikonal equation: φ = n(x), Ansatz only treats one wave. In general crossing waves u(x) A 1 (x)e iωφ 1(x) + A 2 (x)e iωφ 2(x) + Olof Runborg (KTH) High-Frequency Waves INI, / 52

42 Eikonal equation Properties Eikonal equation: φ = n(x), Ansatz only treats one wave. In general crossing waves u(x) A 1 (x)e iωφ 1(x) + A 2 (x)e iωφ 2(x) + Nonlinear equation, no superposition principle Olof Runborg (KTH) High-Frequency Waves INI, / 52

43 Eikonal equation Properties Eikonal equation: φ = n(x), Ansatz only treats one wave. In general crossing waves u(x) A 1 (x)e iωφ1(x) + A 2 (x)e iωφ2(x) + Nonlinear equation, no superposition principle Viscosity solution, kinks Olof Runborg (KTH) High-Frequency Waves INI, / 52

44 Eikonal equation Properties Eikonal equation: φ = n(x), Ansatz only treats one wave. In general crossing waves u(x) A 1 (x)e iωφ1(x) + A 2 (x)e iωφ2(x) + Nonlinear equation, no superposition principle Viscosity solution, kinks First arrival property: φ visc (x) = min n φ n (x) Olof Runborg (KTH) High-Frequency Waves INI, / 52

45 Eikonal equation First arrival property (a) Correct solution (b) Eikonal equation Note: φ traveltime of the wave φ(x) =constant (level sets) represent wave fronts Olof Runborg (KTH) High-Frequency Waves INI, / 52

46 Eikonal equation Example Olof Runborg (KTH) High-Frequency Waves INI, / 52

47 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

48 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

49 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Olof Runborg (KTH) High-Frequency Waves INI, / 52

50 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Ray tracing method of characteristics for eikonal equation. Olof Runborg (KTH) High-Frequency Waves INI, / 52

51 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Ray tracing method of characteristics for eikonal equation. If valid at t = 0, then for all t > 0: φ(x(t)) = p(t), (local ray direction wavefronts) Olof Runborg (KTH) High-Frequency Waves INI, / 52

52 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Ray tracing method of characteristics for eikonal equation. If valid at t = 0, then for all t > 0: φ(x(t)) = p(t), (local ray direction wavefronts) φ(x(t)) = t, (phase traveltime) Olof Runborg (KTH) High-Frequency Waves INI, / 52

53 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Ray tracing method of characteristics for eikonal equation. If valid at t = 0, then for all t > 0: φ(x(t)) = p(t), (local ray direction wavefronts) φ(x(t)) = t, (phase traveltime) p(t) = 1/c(x(t)), (can reduce to p S d 1, in 2D: p θ) Olof Runborg (KTH) High-Frequency Waves INI, / 52

54 Ray tracing Rays are the (bi)characteristics (x(t), p(t)) of the eikonal equation, given by ODEs dx dt = c(x) 2 p, dp dt = c(x) c(x), p(t) is local ray direction, "slowness" vector. Ray tracing method of characteristics for eikonal equation. If valid at t = 0, then for all t > 0: φ(x(t)) = p(t), (local ray direction wavefronts) φ(x(t)) = t, (phase traveltime) p(t) = 1/c(x(t)), (can reduce to p S d 1, in 2D: p θ) There are also ODEs for the amplitude along rays. Olof Runborg (KTH) High-Frequency Waves INI, / 52

55 Ray tracing Numerics Ray equations (with p = 1/c) Methods: dx dt = c(x) 2 p, dp dt = c(x) c(x), Olof Runborg (KTH) High-Frequency Waves INI, / 52

56 Ray tracing Numerics Ray equations (with p = 1/c) dx dt = c(x) 2 p, dp dt = c(x) c(x), Methods: Standard numerical ODE methods, e.g. Runge Kutta. Olof Runborg (KTH) High-Frequency Waves INI, / 52

57 Ray tracing Numerics Ray equations (with p = 1/c) dx dt = c(x) 2 p, dp dt = c(x) c(x), Methods: Standard numerical ODE methods, e.g. Runge Kutta. If c(x) is piecewise constant rays piecewise straight lines refracted/reflected at interfaces by Snell s law. (Geometrical problem, shooting and bouncing rays (SBR).) Olof Runborg (KTH) High-Frequency Waves INI, / 52

58 Ray tracing Numerics Ray equations (with p = 1/c) dx dt = c(x) 2 p, dp dt = c(x) c(x), Methods: Standard numerical ODE methods, e.g. Runge Kutta. If c(x) is piecewise constant rays piecewise straight lines refracted/reflected at interfaces by Snell s law. (Geometrical problem, shooting and bouncing rays (SBR).) Properties: Rays can cross. (No problem with multivaluedness.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

59 Ray tracing Numerics Ray equations (with p = 1/c) dx dt = c(x) 2 p, dp dt = c(x) c(x), Methods: Standard numerical ODE methods, e.g. Runge Kutta. If c(x) is piecewise constant rays piecewise straight lines refracted/reflected at interfaces by Snell s law. (Geometrical problem, shooting and bouncing rays (SBR).) Properties: Rays can cross. (No problem with multivaluedness.) Solution only given along rays. (Hard to interpolate traveltime to regular grid.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

60 Ray tracing Numerics Ray equations (with p = 1/c) dx dt = c(x) 2 p, dp dt = c(x) c(x), Methods: Standard numerical ODE methods, e.g. Runge Kutta. If c(x) is piecewise constant rays piecewise straight lines refracted/reflected at interfaces by Snell s law. (Geometrical problem, shooting and bouncing rays (SBR).) Properties: Rays can cross. (No problem with multivaluedness.) Solution only given along rays. (Hard to interpolate traveltime to regular grid.) Diverging rays. (Hard to cover full domain.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

61 Example: Ray tracing solution Olof Runborg (KTH) High-Frequency Waves INI, / 52

62 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

63 Wavefronts Wavefront given by φ =constant. Suppose initial wavefront is γ(α) and φ(γ) = 0. Let x(t, α), p(t, α) be the rays emitted orthogonal from this curve: x(0, α) = γ(α), p(0, α) = γ (α) c γ (α). Wavefronts also given by: x(t = constant, α). More general definition. Olof Runborg (KTH) High-Frequency Waves INI, / 52

64 Phase space Wavefront in xy-space typically non-smooth. Caustics appear. Olof Runborg (KTH) High-Frequency Waves INI, / 52

65 Phase space Wavefront in xy-space typically non-smooth. Caustics appear. Introduce phase space (x, p), where p S d 1 is local ray direction Olof Runborg (KTH) High-Frequency Waves INI, / 52

66 Phase space Wavefront in xy-space typically non-smooth. Caustics appear. Introduce phase space (x, p), where p S d 1 is local ray direction Observation: Wavefront is a smooth curve in phase space. 2D problems: 1D curve in 3D phase space (x, y, θ). 3D problems: 2D surface in 5D phase space (x, y, z, θ, α). Olof Runborg (KTH) High-Frequency Waves INI, / 52

67 Phase space Wavefront in xy-space typically non-smooth. Caustics appear. Introduce phase space (x, p), where p S d 1 is local ray direction Observation: Wavefront is a smooth curve in phase space. 2D problems: 1D curve in 3D phase space (x, y, θ). 3D problems: 2D surface in 5D phase space (x, y, z, θ, α). Wavefront in phase space sweeps out a smooth surface the Lagrangian submanifold. Olof Runborg (KTH) High-Frequency Waves INI, / 52

68 Wavefront tracking Directly solve for wavefront given by x(t = const, α). Suppose γ(α) is the initial wavefront, x(0, α) = γ(α). Follow ensemble of rays x(t, α) = c 2 p, t x(0, α) = γ(α), p(t, α) = c t c, p(0, α) = γ (α) c γ (α). Note: In principle we do not need to track p. Moving front in normal direction a possibility x t = c x α x α (since 0 = αφ(x(t, α)) = x α φ = x α p) But not good numerically since wavefront non-smooth! Olof Runborg (KTH) High-Frequency Waves INI, / 52

69 Wavefront construction [Vinje, Iversen, Gjöystdal, Lambaré,... ] Solve for x(t, α) and p(t, α). Discretize in α and trace rays for α 1, α 2, α 3,... where α j = j α. x(t n, α j ) Olof Runborg (KTH) High-Frequency Waves INI, / 52

70 Wavefront construction [Vinje, Iversen, Gjöystdal, Lambaré,... ] Solve for x(t, α) and p(t, α). Discretize in α and trace rays for α 1, α 2, α 3,... where α j = j α. Insert new rays adaptively by interpolation when front resolution deteriorates. E.g.: If x(t n, α j+1 ) x(t n, α j ) tol then insert new ray at α j+1/2. x(t n, α j ) Olof Runborg (KTH) High-Frequency Waves INI, / 52

71 Wavefront construction [Vinje, Iversen, Gjöystdal, Lambaré,... ] Solve for x(t, α) and p(t, α). Discretize in α and trace rays for α1, α2, α3,... where αj = j α. Insert new rays adaptively by interpolation when front resolution deteriorates. E.g.: If x(tn, αj+1 ) x(tn, αj ) tol then insert new ray at αj+1/2. Interpolate traveltime/phase/amplitude onto regular grid. x(tn, αj ) Olof Runborg (KTH) High-Frequency Waves INI, / 52

72 Example: Wavefront tracking solution Multiple arrivals ok. Lagrangian method. Interpolation can be complicated. Olof Runborg (KTH) High-Frequency Waves INI, / 52

73 Wavefront tracking, Eulerian versions Level set methods [Osher, Tsai, Cheng, Liu, Jin, Qian,... ] Level set methods: Represent wavefront implicitly as the zero level set of a signed distance function φ(x). γ = {x R d : φ(x) = 0}. Olof Runborg (KTH) High-Frequency Waves INI, / 52

74 Level set methods Wavefront in phase space higher co-dimension: Represent as intersection of zero level sets of several functions. In 2D, γ = {x R 3 : φ 1 (x) = 0, φ 2 (x) = 0}, φ j = φ j (x, y, θ). Level set functions satisfy linear hyperbolic PDEs: t φ j + c(x, y) cos θ x φ j + c(x, y) sin θ y φ j + (c x sin θ c y cos θ) θ φ j = 0. Use local level set method to reduce complexity. Olof Runborg (KTH) High-Frequency Waves INI, / 52

75 Wavefront tracking, Eulerian versions Segment projection method [Engquist, Tornberg, OR] Wavefront γ is given as a union of curve segments γ j. The segments are chosen such that they can be represented by a function of one coordinate variable: f i (t, x) and g j (t, y) From ray equations simple PDEs can be derived for f i and g j. Ex: t f i + u x f i = v. Connectivity of segments is also maintained. (a) Interface (b) x-segments (c) y-segments Olof Runborg (KTH) High-Frequency Waves INI, / 52

76 Segment projection method Example Plane wave entering from the left. (a) Local Ray Directions (b) Wave Fronts Olof Runborg (KTH) High-Frequency Waves INI, / 52

77 Segment projection method Example, segments Olof Runborg (KTH) High-Frequency Waves INI, / 52

78 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

79 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

80 Kinetic formulation View rays as trajectories of particles. Let f (t, x, p) be the particle (photon) density in phase space. Olof Runborg (KTH) High-Frequency Waves INI, / 52

81 Kinetic formulation View rays as trajectories of particles. Let f (t, x, p) be the particle (photon) density in phase space. Bicharacteristic equations Liouville equation f t + c 2 p x f c c pf = 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

82 Kinetic formulation View rays as trajectories of particles. Let f (t, x, p) be the particle (photon) density in phase space. Bicharacteristic equations Liouville equation f t + c 2 p x f c c pf = 0. A ray at x in direction p 0 (t, x) and amplitude A(t, x) represented as f (t, x, p) = A 2 (t, x)δ(p p 0 (t, x)). Olof Runborg (KTH) High-Frequency Waves INI, / 52

83 Kinetic formulation View rays as trajectories of particles. Let f (t, x, p) be the particle (photon) density in phase space. Bicharacteristic equations Liouville equation f t + c 2 p x f c c pf = 0. A ray at x in direction p 0 (t, x) and amplitude A(t, x) represented as f (t, x, p) = A 2 (t, x)δ(p p 0 (t, x)). Since p = n(x), the density f supported on sphere p = n(x). Olof Runborg (KTH) High-Frequency Waves INI, / 52

84 Kinetic formulation Relation to Wigner theory Liouville equation can also be derived directly from wave eq. through e.g. Wigner measures [Tartar, Lions, Paul, Gerard, Mauser, Markowich, Poupaud,... ] Olof Runborg (KTH) High-Frequency Waves INI, / 52

85 Kinetic formulation Relation to Wigner theory Liouville equation can also be derived directly from wave eq. through e.g. Wigner measures [Tartar, Lions, Paul, Gerard, Mauser, Markowich, Poupaud,... ] Let f be limit of the Wigner tranform of Helmholtz solution f = lim ω F y pu(x + y/2ω)u(x y/2ω). Then f satisfies Liouville eq. f t + c 2 p x f c c pf = 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

86 Kinetic formulation Relation to Wigner theory Liouville equation can also be derived directly from wave eq. through e.g. Wigner measures [Tartar, Lions, Paul, Gerard, Mauser, Markowich, Poupaud,... ] Let f be limit of the Wigner tranform of Helmholtz solution f = lim ω F y pu(x + y/2ω)u(x y/2ω). Then f satisfies Liouville eq. f t + c 2 p x f c c pf = 0. Relationship to wave equation solution: u = Ae iωφ f = A 2 δ (p φ). Olof Runborg (KTH) High-Frequency Waves INI, / 52

87 Kinetic formulation Relation to Wigner theory Liouville equation can also be derived directly from wave eq. through e.g. Wigner measures [Tartar, Lions, Paul, Gerard, Mauser, Markowich, Poupaud,... ] Let f be limit of the Wigner tranform of Helmholtz solution f = lim ω F y pu(x + y/2ω)u(x y/2ω). Then f satisfies Liouville eq. f t + c 2 p x f c c pf = 0. Relationship to wave equation solution: u = Ae iωφ f = A 2 δ (p φ). lim F y pa(x + y/2ω)a(x y/2ω)e iω(φ(x+y/2ω) φ(x y/2ω)) ω = F y p A 2 (x)e iy φ = A 2 δ (p φ). Olof Runborg (KTH) High-Frequency Waves INI, / 52

88 Kinetic formulation Relation to Wigner theory Liouville equation can also be derived directly from wave eq. through e.g. Wigner measures [Tartar, Lions, Paul, Gerard, Mauser, Markowich, Poupaud,... ] Let f be limit of the Wigner tranform of Helmholtz solution f = lim ω F y pu(x + y/2ω)u(x y/2ω). Then f satisfies Liouville eq. f t + c 2 p x f c c pf = 0. Relationship to wave equation solution: u = Ae iωφ f = A 2 δ (p φ). Note: Loss of phase information. Olof Runborg (KTH) High-Frequency Waves INI, / 52

89 Moment equations Derived from Liouville equation in phase space + closure assumption for a system of equations representing the moments. (C.f. hydrodynamic limit from Boltzmann eq.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

90 Moment equations Derived from Liouville equation in phase space + closure assumption for a system of equations representing the moments. (C.f. hydrodynamic limit from Boltzmann eq.) Full equation for f expensive to solve numerically. 6 independent variables in 3D. Olof Runborg (KTH) High-Frequency Waves INI, / 52

91 Moment equations Derived from Liouville equation in phase space + closure assumption for a system of equations representing the moments. (C.f. hydrodynamic limit from Boltzmann eq.) Full equation for f expensive to solve numerically. 6 independent variables in 3D. Moment eq is a PDE description in the small (t, x)-space. Fixed Eulerian grids can be used. Olof Runborg (KTH) High-Frequency Waves INI, / 52

92 Moment equations Derived from Liouville equation in phase space + closure assumption for a system of equations representing the moments. (C.f. hydrodynamic limit from Boltzmann eq.) Full equation for f expensive to solve numerically. 6 independent variables in 3D. Moment eq is a PDE description in the small (t, x)-space. Fixed Eulerian grids can be used. Arbitrary good superposition. N crossing waves allowed. (But larger N means a larger system of PDEs must be solved.) [Brenier, Corrias, Engquist, OR] (wave equation), [Gosse, Jin, Li, Markowich, Sparber] (Schrödinger) Olof Runborg (KTH) High-Frequency Waves INI, / 52

93 Moment equations Derivations, homogenous case (c 1) Starting point is f t + p x f = 0. Let p = (p 1, p 2 ). Define the moments, m ij = p1 i pj R 2 2 fdp. From p1 i pj R 2 2 (f t + p x f )dp = 0, we get the infinite (valid i, j 0) system of moment equations (m ij ) t + (m i+1,j ) x + (m i,j+1 ) y = 0. Olof Runborg (KTH) High-Frequency Waves INI, / 52

94 Moment equations Derivations, homogenous case, cont. Make the closure assumption N f (x, p, t) = A 2 k δ(p p k ), k=1 ( ) cos p k = θk. sin θ k The moments take the form m ij = N A 2 k cosi θ k sin j θ k. k=1 Corresponds to a maximum of N waves at each point. Choose equations for moments m 2k 1,0 and m 0,2k 1, k = 1,..., N. Gives closed system of 2N equations with 2N unknowns (the A k s and θ k s). Olof Runborg (KTH) High-Frequency Waves INI, / 52

95 Moment equations Examples Ex. N = 1 ( u1 u 2 ) t + u 1 2 u 2 1 +u2 2 u 1 u 2 u 2 1 +u2 2 x + u 1 u 2 u 2 1 +u2 2 u 2 2 u 2 1 +u2 2 y = 0. where u 1 = m 10 = A 2 cos θ and u 2 = m 01 = A 2 sin θ. For N 2, F 0 (u) t + F 1 (u) x + F 2 (u) y = 0. where F 0 (u), F 1 (u) and F 2 (u) are complicated non-linear functions. PDE = weakly hyperbolic system of conservation laws, (with source terms when c varies) Flux functions in conservation law can be difficult to evaluate. Olof Runborg (KTH) High-Frequency Waves INI, / 52

96 Moment equations Wedge example (a) N = 1 (b) N = 2 Olof Runborg (KTH) High-Frequency Waves INI, / 52

97 Geometrical optics models and numerical methods u + ω 2 n(x) 2 u = 0 Rays dx dt = c 2 p, dp dt Ray tracing = c c Wavefront methods Kinetic f t + c 2 p x f 1 c c pf = 0 Moment methods, Phase space methods Eikonal φ = n(x) Hamilton Jacobi methods Olof Runborg (KTH) High-Frequency Waves INI, / 52

98 Phase Space Methods Eikonal solvers, wavefront tracking, etc. essentially constructed to solve the problem for one single initial data, e.g. a point source or a plane wave. Solves for a "sheet" in phase space (the Lagrangian submanifold). Some applications demand solution for many different sets of initial data (e.g. inverse problems). Phase space methods: Get solution in the whole phase space. (Corresponds to solving for all possible initial data.) Olof Runborg (KTH) High-Frequency Waves INI, / 52

99 Phase Space Methods, cont. Equation to solve typically of the type time-independent Liouville equation/whole level set equation. (But other interpretation of unknown.) Computational cost: Suppose N discretization points in each dimension. In 2D eikonal solvers/wavefront construction typically cost O(N 2 ) for one initial data. Phase space methods cost O(N 3 log N) for all initial data. Combining advantages of using a fixed grid (cf eikonal solvers) and capturing multiple arrivals (cf ray methods) relatively easy. Examples: Phase Flow Method [Candès, Ying], Fast phase space method [Fomel, Sethian]. Olof Runborg (KTH) High-Frequency Waves INI, / 52

100 Physical optics Integral formulation of scattering problem u s (x) = G( x x ) u(x ) n dx, x R d \ Ω, Ω Approximate u n by geometrical optics solution. E.g. if u inc = exp(iωα x) is a plane wave, Ω convex, then { u n (u inc + us GO ) 2iωα ˆn(x)e iωα x, x illuminated, = n 0, x in shadow. Cost of evaluating solution still depends on ω. Olof Runborg (KTH) High-Frequency Waves INI, / 52

101 Physical optics Exact physical optics For convex Ω, make ansatz u n = iωa(x, ω)eiωα x then A(x, ω) smooth, uniformly in ω, except at shadow boundaries. Discretize and solve A(x, ω) at cost independent of ω. [Bruno, Chandler Wilde,... ] Olof Runborg (KTH) High-Frequency Waves INI, / 52

102 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Olof Runborg (KTH) High-Frequency Waves INI, / 52

103 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Olof Runborg (KTH) High-Frequency Waves INI, / 52

104 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Trade-off point in ω increases with increasing computer power and is problem and accuracy dependent. Olof Runborg (KTH) High-Frequency Waves INI, / 52

105 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Trade-off point in ω increases with increasing computer power and is problem and accuracy dependent. Challenges for geometrical optics methods include capturing multiple arrivals on fixed grids at reasonable complexity. Olof Runborg (KTH) High-Frequency Waves INI, / 52

106 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Trade-off point in ω increases with increasing computer power and is problem and accuracy dependent. Challenges for geometrical optics methods include capturing multiple arrivals on fixed grids at reasonable complexity. Phase space methods useful when many similar problems need to be solved. Olof Runborg (KTH) High-Frequency Waves INI, / 52

107 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Trade-off point in ω increases with increasing computer power and is problem and accuracy dependent. Challenges for geometrical optics methods include capturing multiple arrivals on fixed grids at reasonable complexity. Phase space methods useful when many similar problems need to be solved. Hybrid methods coupling elements of asymptotic models and direct methods a possibility. Olof Runborg (KTH) High-Frequency Waves INI, / 52

108 Concluding remarks Complexity of simulation based on high-frequency approximations are not ω-dependent. Error normally O(1/ω). Boundaries and caustic cause bigger errors. GTD reduces them. Trade-off point in ω increases with increasing computer power and is problem and accuracy dependent. Challenges for geometrical optics methods include capturing multiple arrivals on fixed grids at reasonable complexity. Phase space methods useful when many similar problems need to be solved. Hybrid methods coupling elements of asymptotic models and direct methods a possibility. Methods generalize to Maxwell, elastic wave eq, etc. Olof Runborg (KTH) High-Frequency Waves INI, / 52

109 References B. Engquist and O. Runborg. Computational high-frequency wave propagation. Acta Numerica, 12: , J.-D. Benamou. An introduction to Eulerian geometrical optics ( ). J. Sci. Comput., 19(1-3):63 93, O. Runborg. Mathematical models and numerical methods for high frequency waves. To appear in Commun. Comp. Phys., Olof Runborg (KTH) High-Frequency Waves INI, / 52