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annotate src/SbpOperators/volumeops/laplace/laplace.jl @ 1106:b4ee47f2aafb feature/boundary_conditions

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Start implementing functions for boundary conditions associated with Laplace
author Vidar Stiernström <vidar.stiernstrom@it.uu.se>
date Mon, 13 Jun 2022 13:46:24 +0200
parents 7fc8df5157a7
children 302d36b5ba8e
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1 """
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2 Laplace{T, Dim, TM} <: LazyTensor{T, Dim, Dim}
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4 Implements the Laplace operator, approximating ∑d²/xᵢ² , i = 1,...,`Dim` as a
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5 `LazyTensor`. Additionally `Laplace` stores the `StencilSet`
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6 used to construct the `LazyTensor `.
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7 """
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8 struct Laplace{T, Dim, TM<:LazyTensor{T, Dim, Dim}} <: LazyTensor{T, Dim, Dim}
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9 D::TM # Difference operator
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10 stencil_set::StencilSet # Stencil set of the operator
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11 end
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12
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13 """
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14 Laplace(grid::Equidistant, stencil_set)
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15
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16 Creates the `Laplace` operator `Δ` on `grid` given a `stencil_set`.
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17
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18 See also [`laplace`](@ref).
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19 """
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20 function Laplace(grid::EquidistantGrid, stencil_set::StencilSet)
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21 inner_stencil = parse_stencil(stencil_set["D2"]["inner_stencil"])
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22 closure_stencils = parse_stencil.(stencil_set["D2"]["closure_stencils"])
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23 Δ = laplace(grid, inner_stencil,closure_stencils)
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24 return Laplace(Δ,stencil_set)
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25 end
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27 LazyTensors.range_size(L::Laplace) = LazyTensors.range_size(L.D)
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28 LazyTensors.domain_size(L::Laplace) = LazyTensors.domain_size(L.D)
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29 LazyTensors.apply(L::Laplace, v::AbstractArray, I...) = LazyTensors.apply(L.D,v,I...)
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30
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31 # TODO: Implement pretty printing of Laplace once pretty printing of LazyTensors is implemented.
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32 # Base.show(io::IO, L::Laplace) = ...
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34 """
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35 laplace(grid::EquidistantGrid, inner_stencil, closure_stencils)
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36
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37 Creates the Laplace operator operator `Δ` as a `LazyTensor`
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39 `Δ` approximates the Laplace operator ∑d²/xᵢ² , i = 1,...,`Dim` on `grid`, using
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40 the stencil `inner_stencil` in the interior and a set of stencils `closure_stencils`
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41 for the points in the closure regions.
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42
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43 On a one-dimensional `grid`, `Δ` is equivalent to `second_derivative`. On a
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44 multi-dimensional `grid`, `Δ` is the sum of multi-dimensional `second_derivative`s
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45 where the sum is carried out lazily.
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47 See also: [`second_derivative`](@ref).
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48 """
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49 function laplace(grid::EquidistantGrid, inner_stencil, closure_stencils)
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50 Δ = second_derivative(grid, inner_stencil, closure_stencils, 1)
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51 for d = 2:dimension(grid)
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52 Δ += second_derivative(grid, inner_stencil, closure_stencils, d)
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53 end
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54 return Δ
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55 end
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57 sat(Δ::Laplace, g::EquidistantGrid, bc::BoundaryCondition{Neumann}) = neumann_sat(Δ, g, bc.id)
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58
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59 function neumann_sat(Δ::Laplace, g::EquidistantGrid{1}, id::BoundaryIdentifier)
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60 set = Δ.stencil_set
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61 H⁻¹ = inverse_inner_product(g,set)
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62 e = boundary_restriction(g, set, id)
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63 d = normal_derivative(g, set, id)
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64 return f(u,data) = H⁻¹∘e'∘(d*u .- data)
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65 #closure = H⁻¹∘e'∘d
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66 #penalty = -H⁻¹∘e'
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67 #return closure, penalty
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68 end
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72 function neumann_sat(Δ::Laplace, g::EquidistantGrid, id::BoundaryIdentifier)
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73 set = Δ.stencil_set
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74 H⁻¹ = inverse_inner_product(g, set)
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75 orth_dims = Tuple(filter(i -> i != dimension(g), 1:dimension(g)))
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76 Hᵧ = inner_product(restrict(g, orth_dims...), set)
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77 e = boundary_restriction(g, set, id)
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78 d = normal_derivative(g, set, id)
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79 return f(u,data) = H⁻¹∘e'∘Hᵧ∘(d*u .- data)
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80 #closure = H⁻¹∘e'∘Hᵧ∘d
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81 #penalty = -H⁻¹∘e'∘Hᵧ
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82 #return closure, penalty
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83 end