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comparison src/SbpOperators/volumeops/laplace/laplace.jl @ 1854:654a2b7e6824 tooling/benchmarks

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author Jonatan Werpers <jonatan@werpers.com>
date Sat, 11 Jan 2025 10:19:47 +0100
parents b5690ab5f0b8
children f3d7e2d7a43f
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1378:2b5480e2d4bf 1854:654a2b7e6824
50 Δ += LazyTensors.inflate(laplace(g.grids[d], stencil_set), size(g), d) 50 Δ += LazyTensors.inflate(laplace(g.grids[d], stencil_set), size(g), d)
51 end 51 end
52 return Δ 52 return Δ
53 end 53 end
54 laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set) 54 laplace(g::EquidistantGrid, stencil_set) = second_derivative(g, stencil_set)
55
56 """
57 sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning, R_tuning)
58
59 The operators required to construct the SAT for imposing a Dirichlet
60 condition. `H_tuning` and `R_tuning` are used to specify the strength of the
61 penalty.
62
63 See also: [`sat`](@ref), [`DirichletCondition`](@ref), [`positivity_decomposition`](@ref).
64 """
65 function sat_tensors(Δ::Laplace, g::Grid, bc::DirichletCondition; H_tuning = 1., R_tuning = 1.)
66 id = boundary(bc)
67 set = Δ.stencil_set
68 H⁻¹ = inverse_inner_product(g,set)
69 Hᵧ = inner_product(boundary_grid(g, id), set)
70 e = boundary_restriction(g, set, id)
71 d = normal_derivative(g, set, id)
72 B = positivity_decomposition(Δ, g, boundary(bc); H_tuning, R_tuning)
73 penalty_tensor = H⁻¹∘(d' - B*e')∘Hᵧ
74 return penalty_tensor, e
75 end
76
77 """
78 sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
79
80 The operators required to construct the SAT for imposing a Neumann condition.
81
82 See also: [`sat`](@ref), [`NeumannCondition`](@ref).
83 """
84 function sat_tensors(Δ::Laplace, g::Grid, bc::NeumannCondition)
85 id = boundary(bc)
86 set = Δ.stencil_set
87 H⁻¹ = inverse_inner_product(g,set)
88 Hᵧ = inner_product(boundary_grid(g, id), set)
89 e = boundary_restriction(g, set, id)
90 d = normal_derivative(g, set, id)
91
92 penalty_tensor = -H⁻¹∘e'∘Hᵧ
93 return penalty_tensor, d
94 end
95
96 """
97 positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
98
99 Constructs the scalar `B` such that `d' - 1/2*B*e'` is symmetric positive
100 definite with respect to the boundary quadrature. Here `d` is the normal
101 derivative and `e` is the boundary restriction operator. `B` can then be used
102 to form a symmetric and energy stable penalty for a Dirichlet condition. The
103 parameters `H_tuning` and `R_tuning` are used to specify the strength of the
104 penalty and must be greater than 1. For details we refer to
105 <https://doi.org/10.1016/j.jcp.2020.109294>
106 """
107 function positivity_decomposition(Δ::Laplace, g::Grid, b::BoundaryIdentifier; H_tuning, R_tuning)
108 @assert(H_tuning ≥ 1.)
109 @assert(R_tuning ≥ 1.)
110 Nτ_H, τ_R = positivity_limits(Δ,g,b)
111 return H_tuning*Nτ_H + R_tuning*τ_R
112 end
113
114 function positivity_limits(Δ::Laplace, g::EquidistantGrid, b::BoundaryIdentifier)
115 h = spacing(g)
116 θ_H = parse_scalar(Δ.stencil_set["H"]["closure"][1])
117 θ_R = parse_scalar(Δ.stencil_set["D2"]["positivity"]["theta_R"])
118
119 τ_H = one(eltype(Δ))/(h*θ_H)
120 τ_R = one(eltype(Δ))/(h*θ_R)
121 return τ_H, τ_R
122 end
123
124 function positivity_limits(Δ::Laplace, g::TensorGrid, b::BoundaryIdentifier)
125 τ_H, τ_R = positivity_limits(Δ, g.grids[grid_id(b)], b)
126 return τ_H*ndims(g), τ_R
127 end