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annotate src/SbpOperators/volumeops/laplace/laplace.jl @ 1619:1937be9502a7 feature/boundary_conditions

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REVIEW: Minor fixes to doc strings in laplace.jl
author Jonatan Werpers <jonatan@werpers.com>
date Tue, 11 Jun 2024 00:19:09 +0200
parents e41eddc640f3
children 707fc9761c2b f28c92ec843c 84aed3abab94
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1 """
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2 Laplace{T, Dim, TM} <: LazyTensor{T, Dim, Dim}
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3
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4 The Laplace operator, approximating ∑d²/xᵢ² , i = 1,...,`Dim` as a
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5 `LazyTensor`.
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6 """
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7 struct Laplace{T, Dim, TM<:LazyTensor{T, Dim, Dim}} <: LazyTensor{T, Dim, Dim}
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8 D::TM # Difference operator
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9 stencil_set::StencilSet # Stencil set of the operator
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10 end
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11
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12 """
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13 Laplace(g::Grid, stencil_set::StencilSet)
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14
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15 Creates the `Laplace` operator `Δ` on `g` given `stencil_set`.
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16
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17 See also [`laplace`](@ref).
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18 """
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19 function Laplace(g::Grid, stencil_set::StencilSet)
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20 Δ = laplace(g, stencil_set)
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21 return Laplace(Δ, stencil_set)
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22 end
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23
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24 LazyTensors.range_size(L::Laplace) = LazyTensors.range_size(L.D)
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25 LazyTensors.domain_size(L::Laplace) = LazyTensors.domain_size(L.D)
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26 LazyTensors.apply(L::Laplace, v::AbstractArray, I...) = LazyTensors.apply(L.D,v,I...)
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27
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28 # TODO: Implement pretty printing of Laplace once pretty printing of LazyTensors is implemented.
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29 # Base.show(io::IO, L::Laplace) = ...
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31 """
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32 laplace(g::Grid, stencil_set)
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33
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34 Creates the Laplace operator operator `Δ` as a `LazyTensor` on `g`.
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35
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36 `Δ` approximates the Laplace operator ∑d²/xᵢ² , i = 1,...,`Dim` on `g`. The
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37 approximation depends on the type of grid and the stencil set.
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38
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39 See also: [`second_derivative`](@ref).
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40 """
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41 function laplace end
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42 function laplace(g::TensorGrid, stencil_set)
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43 # return mapreduce(+, enumerate(g.grids)) do (i, gᵢ)
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44 # Δᵢ = laplace(gᵢ, stencil_set)
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45 # LazyTensors.inflate(Δᵢ, size(g), i)
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46 # end
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47
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48 Δ = LazyTensors.inflate(laplace(g.grids[1], stencil_set), size(g), 1)
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49 for d = 2:ndims(g)
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50 Δ += LazyTensors.inflate(laplace(g.grids[d], stencil_set), size(g), d)
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51 end
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52 return Δ
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53 end
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54 laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set)
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55
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56 """
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57 sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
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58
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59 The operators required to construct the SAT for imposing a Dirichlet
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60 condition. `H_tuning` and `R_tuning` are used to specify the strength of the
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61 penalty.
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62
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63 See also: [`sat`](@ref),[`DirichletCondition`](@ref), [`positivity_decomposition`](@ref).
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64 """
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65 function sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning = 1., R_tuning = 1.)
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66 id = boundary(bc)
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67 set = Δ.stencil_set
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68 H⁻¹ = inverse_inner_product(g,set)
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69 Hᵧ = inner_product(boundary_grid(g, id), set)
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70 e = boundary_restriction(g, set, id)
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71 d = normal_derivative(g, set, id)
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72 B = positivity_decomposition(Δ, g, bc; H_tuning, R_tuning)
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73 penalty_tensor = H⁻¹∘(d' - B*e')∘Hᵧ
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74 return penalty_tensor, e
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75 end
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76
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77 """
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78 sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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79
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80 The operators required to construct the SAT for imposing a Neumann condition.
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81
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82 See also: [`sat`](@ref), [`NeumannCondition`](@ref).
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83 """
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84 function sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
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85 id = boundary(bc)
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86 set = Δ.stencil_set
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87 H⁻¹ = inverse_inner_product(g,set)
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88 Hᵧ = inner_product(boundary_grid(g, id), set)
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89 e = boundary_restriction(g, set, id)
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90 d = normal_derivative(g, set, id)
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91
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92 penalty_tensor = -H⁻¹∘e'∘Hᵧ
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93 return penalty_tensor, d
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94 end
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95
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96 """
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97 positivity_decomposition(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
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98
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99 Constructs the scalar `B` such that `d' - 1/2*B*e'` is symmetric positive
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100 definite with respect to the boundary quadrature. Here `d` is the normal
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101 derivative and `e` is the boundary restriction operator. `B` can then be used
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102 to form a symmetric and energy stable penalty for a Dirichlet condition. The
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103 parameters `H_tuning` and `R_tuning` are used to specify the strength of the
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104 penalty and must be greater than 1. For details we refer to
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105 https://doi.org/10.1016/j.jcp.2020.109294
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106 """
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107 function positivity_decomposition(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
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108 @assert(H_tuning ≥ 1.)
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109 @assert(R_tuning ≥ 1.)
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110 Nτ_H, τ_R = positivity_limits(Δ,g,bc)
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111 return H_tuning*Nτ_H + R_tuning*τ_R
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112 end
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113
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114 # TODO: We should consider implementing a proper BoundaryIdentifier for EquidistantGrid and then
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115 # change bc::BoundaryCondition to id::BoundaryIdentifier
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116 function positivity_limits(Δ::Laplace, g::EquidistantGrid, bc::DirichletCondition)
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117 h = spacing(g)
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118 θ_H = parse_scalar(Δ.stencil_set["H"]["closure"][1])
8315c456e3b4 Simplify parsing of constants from stencil set
Jonatan Werpers <jonatan@werpers.com>
parents: 1607
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119 θ_R = parse_scalar(Δ.stencil_set["D2"]["positivity"]["theta_R"])
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parents: 1607
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120
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7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
Jonatan Werpers <jonatan@werpers.com>
parents: 1606
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121 τ_H = 1/(h*θ_H)
7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
Jonatan Werpers <jonatan@werpers.com>
parents: 1606
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122 τ_R = 1/(h*θ_R)
7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
Jonatan Werpers <jonatan@werpers.com>
parents: 1606
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123 return τ_H, τ_R
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19cdec9c21cb Implement and test sat_tensors for Dirichlet and Neumann conditions
Vidar Stiernström <vidar.stiernstrom@gmail.com>
parents: 1484
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124 end
19cdec9c21cb Implement and test sat_tensors for Dirichlet and Neumann conditions
Vidar Stiernström <vidar.stiernstrom@gmail.com>
parents: 1484
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125
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7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
Jonatan Werpers <jonatan@werpers.com>
parents: 1606
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126 function positivity_limits(Δ::Laplace, g::TensorGrid, bc::DirichletCondition)
7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
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parents: 1606
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127 τ_H, τ_R = positivity_limits(Δ, g.grids[grid_id(boundary(bc))], bc)
7216448d0c5a REVIEW: Suggest deduplication of positivity decompostion code
Jonatan Werpers <jonatan@werpers.com>
parents: 1606
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128 return τ_H*ndims(g), τ_R
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3e7438e2a033 Address review comments (1 left to be discussed)
Vidar Stiernström <vidar.stiernstrom@gmail.com>
parents: 1600
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129 end