Add subcell finite volume differencing and reconstruction operators

src/RefElemData_SBP.jl

@@ -194,158 +194,5 @@ function hybridized_SBP_operators(rd)
     return Qrsth, VhP, Ph, Vh
 end
 
-# default to doing nothing
-map_nodes_to_symmetric_element(element_type, rst...) = rst
 
-# for triangles, map to an equilateral triangle
-function map_nodes_to_symmetric_element(::Tri, r, s)
-    # biunit right triangular vertices
-    v1, v2, v3 = SVector{2}.(zip(nodes(Tri(), 1)...))
 
-    denom = (v2[2] - v3[2]) * (v1[1] - v3[1]) + (v3[1] - v2[1]) * (v1[2] - v3[2])
-    L1 = @. ((v2[2] - v3[2]) * (r - v3[1]) + (v3[1] - v2[1]) * (s - v3[2])) / denom
-    L2 = @. ((v3[2] - v1[2]) * (r - v3[1]) + (v1[1] - v3[1]) * (s - v3[2])) / denom
-    L3 = @. 1 - L1 - L2
-
-    # equilateral vertices
-    v1 = SVector{2}(2 * [-.5, -sqrt(3) / 6])
-    v2 = SVector{2}(2 * [.5, -sqrt(3)/6])
-    v3 = SVector{2}(2 * [0, sqrt(3)/3])
-
-    x = @. v1[1] * L1 + v2[1] * L2 + v3[1] * L3
-    y = @. v1[2] * L1 + v2[2] * L2 + v3[2] * L3
-
-    return x, y
-end
-
-
-"""
-    function sparse_low_order_SBP_operators(rd)
-
-Constructs sparse low order SBP operators given a `RefElemData`. 
-Returns operators `Qrst..., E ≈ Vf * Pq` that satisfy a generalized 
-summation-by-parts (GSBP) property:
-
-        `Q_i + Q_i^T = E' * B_i * E`
-"""
-function sparse_low_order_SBP_operators(rd::RefElemData{NDIMS}) where {NDIMS}
-    (; Pq, Vf, rstq, wf, nrstJ) = rd
-
-    # if element is a simplex, convert nodes to an equilateral triangle for symmetry
-    rstq = map_nodes_to_symmetric_element(rd.element_type, rstq...)
-
-    # build distance and adjacency matrices
-    D = [norm(SVector{NDIMS}(getindex.(rstq, i)) - SVector{NDIMS}(getindex.(rstq, j))) 
-            for i in eachindex(first(rstq)), j in eachindex(first(rstq))]
-    A = zeros(Int, length(first(rstq)), length(first(rstq)))
-    for i in axes(D, 1)
-        # heuristic cutoff - we take NDIMS + 1 neighbors, but the smallest distance = 0, 
-        # so we need to access the first NDIMS + 2 sorted distance entries.
-        dist = sort(view(D, i, :))[NDIMS + 2] 
-        for j in findall(view(D, i, :) .< 1.01 * dist)
-            A[i, j] = 1
-        end
-    end
-    A = (A + A' .> 0) # symmetrize
-    L = Diagonal(vec(sum(A, dims=2))) - A
-    sorted_eigvals = sort(eigvals(L))
-
-    # check that there's only one zero null vector
-    @assert sorted_eigvals[2] > 100 * eps() 
-
-    E_dense = Vf * Pq
-    E = zeros(size(E_dense))
-    for i in axes(E, 1)
-        # find all j such that E[i,j] ≥ 0.5, e.g., points which positively contribute to at least half of the 
-        # interpolation. These seem to be associated with volume points "j" that are close to face point "i".
-        ids = findall(E_dense[i, :] .>= 0.5)
-        E[i, ids] .= E_dense[i, ids] ./ sum(E_dense[i, ids]) # normalize so sum(E, dims=2) = [1, 1, ...] still.
-    end
-    Brst = (nJ -> diagm(wf .* nJ)).(nrstJ)
-
-    # compute potential 
-    e = ones(size(L, 2))
-    right_hand_sides = map(B -> 0.5 * sum(E' * B, dims=2), Brst)
-    psi_augmented = map(b -> [L e; e' 0] \ [b; 0], right_hand_sides)
-    psi = map(x -> x[1:end-1], psi_augmented)
-
-    # construct sparse skew part
-    function construct_skew_matrix_from_potential(ψ)
-        S = zeros(length(ψ), length(ψ))
-        for i in axes(S, 1), j in axes(S, 2)
-            if A[i,j] > 0
-                S[i,j] = ψ[j] - ψ[i]
-            end
-        end
-        return S
-    end
-
-    Srst = construct_skew_matrix_from_potential.(psi)
-    Qrst = map((S, B) -> S + 0.5 * E' * B * E, Srst, Brst)
-    return sparse.(Qrst), sparse(E)
-end
-
-function sparse_low_order_SBP_1D_operators(rd::RefElemData) 
-    E = zeros(2, rd.N+1)
-    E[1, 1] = 1
-    E[2, end] = 1
-
-    # create volume operators
-    Q =  diagm(1 => ones(rd.N), -1 => -ones(rd.N))
-    Q[1,1] = -1
-    Q[end,end] = 1
-    Q = 0.5 * Q
-
-    return (sparse(Q),), sparse(E)
-end
-
-sparse_low_order_SBP_operators(rd::RefElemData{1, Line, <:Union{<:SBP, <:Polynomial{Gauss}}}) = 
-    sparse_low_order_SBP_1D_operators(rd)
-
-function diagonal_1D_mass_matrix(N, ::SBP)
-    _, w1D = gauss_lobatto_quad(0, 0, N)
-    return Diagonal(w1D)
-end
-
-function diagonal_1D_mass_matrix(N, ::Polynomial{Gauss})
-    _, w1D = gauss_quad(0, 0, N)
-    return Diagonal(w1D)
-end
-
-function sparse_low_order_SBP_operators(rd::RefElemData{2, Quad, <:Union{<:SBP, <:Polynomial{Gauss}}}) 
-    (Q1D,), E1D = sparse_low_order_SBP_1D_operators(rd)
-
-    # permute face node ordering for the first 2 faces
-    ids = reshape(1:(rd.N+1) * 2, :, 2)
-    Er = zeros((rd.N+1) * 2, rd.Np)
-    Er[vec(ids'), :] .= kron(I(rd.N+1), E1D)
-    Es = kron(E1D, I(rd.N+1))
-    E = vcat(Er, Es)
-
-    M1D = diagonal_1D_mass_matrix(rd.N, rd.approximation_type)
-    Qr = kron(M1D, Q1D)
-    Qs = kron(Q1D, M1D)
-
-    return sparse.((Qr, Qs)), sparse(E)
-end
-
-function sparse_low_order_SBP_operators(rd::RefElemData{3, Hex, <:Union{<:SBP, <:Polynomial{Gauss}}}) 
-    (Q1D,), E1D = sparse_low_order_SBP_1D_operators(rd)
-
-    # permute face nodes for the faces in the ±r and ±s directions
-    ids = reshape(1:(rd.N+1)^2 * 2, rd.N+1, :, 2) 
-    Er, Es, Et = ntuple(_ -> zeros((rd.N+1)^2 * 2, rd.Np), 3)
-    Er[vec(permutedims(ids, [2, 3, 1])), :] .= kron(I(rd.N+1), E1D, I(rd.N+1))
-    Es[vec(permutedims(ids, [3, 2, 1])), :] .= kron(I(rd.N+1), I(rd.N+1), E1D)
-    Et .= kron(E1D, I(rd.N+1), I(rd.N+1))
-
-    # create boundary extraction matrix
-    E = vcat(Er, Es, Et)
-
-    M1D = diagonal_1D_mass_matrix(rd.N, rd.approximation_type)
-    Qr = kron(M1D, Q1D, M1D)
-    Qs = kron(M1D, M1D, Q1D)
-    Qt = kron(Q1D, M1D, M1D)
-
-    return sparse.((Qr, Qs, Qt)), sparse(E)
-end

src/StartUpDG.jl

@@ -7,7 +7,7 @@ using FillArrays: Fill
 using HDF5: h5open # used to read in SBP triangular node data
 using Kronecker: kronecker # for Hex element matrix manipulations
 using LinearAlgebra:
-    cond, diagm, eigvals, Diagonal, UniformScaling, I, mul!, norm, qr, ColumnNorm, Symmetric
+    cond, diagm, eigvals, Diagonal, UniformScaling, I, mul!, norm, qr, ColumnNorm, Symmetric, nullspace, pinv
 using NodesAndModes: meshgrid, find_face_nodes, face_vertices
 @reexport using NodesAndModes # for basis functions
 using PathIntersections: PathIntersections
@@ -17,13 +17,12 @@ using RecipesBase: RecipesBase
 @reexport using RecursiveArrayTools: NamedArrayPartition
 using StaticArrays: SVector, SMatrix
 using Setfield: setproperties, @set # for "modifying" structs (setproperties)
-using SparseArrays: sparse, droptol!, blockdiag
+using SparseArrays: sparse, droptol!, blockdiag, nnz
 using Triangulate: Triangulate, TriangulateIO, triangulate
 @reexport using WriteVTK
 
 @inline mean(x) = sum(x) / length(x)
 
-# reference element utility functions
 include("RefElemData.jl")
 
 include("RefElemData_polynomial.jl")
@@ -37,7 +36,11 @@ export TensorProductWedge
 include("RefElemData_SBP.jl")
 export SBP, DefaultSBPType, TensorProductLobatto, Hicken, Kubatko # types for SBP node dispatch
 export LobattoFaceNodes, LegendreFaceNodes # type parameters for SBP{Kubatko{...}}
-export hybridized_SBP_operators, sparse_low_order_SBP_operators
+export hybridized_SBP_operators
+
+include("low_order_sbp.jl")
+export sparse_low_order_SBP_operators
+export subcell_limiting_operators
 export inverse_trace_constant, face_type
 
 include("ref_elem_utils.jl")