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July 5, 2017 07:23
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function literature_example() | |
# Problem: Δu + 1000uₓ = f | |
# u = 0 on the boundaries | |
# f(x, y, z) = exp(xyz) sin(πx) sin(πy) sin(πz) | |
# 2nd order central differences (shows serious wiggles) | |
# Unknowns per dimension | |
N = 50 | |
# Total number of unknowns | |
n = N^3 | |
# Mesh width | |
h = 1.0 / (N + 1) | |
# Interior points only | |
xs = linspace(0, 1, N + 2)[2 : N + 1] | |
# The Laplacian | |
Δ = poisson_matrix(Float64, N, 3) ./ -h^2 | |
# And the dx bit. | |
∂x_1d = spdiagm((fill(-1000.0 / 2h, N - 1), fill(1000.0 / 2h, N - 1)), (-1, 1)) | |
∂x = kron(speye(N^2), ∂x_1d) | |
# Final matrix and rhs. | |
A = Δ + ∂x | |
b = reshape([f(x, y, z) for x ∈ xs, y ∈ xs, z ∈ xs], n) | |
x, res = bicgstabl(A, b, 2, maxiter = 500, tol = eps()) | |
# @time xs, reshape(A \ rhs, N, N, N) | |
end | |
function poisson_matrix{T}(::Type{T}, n, dims) | |
D = second_order_central_diff(T, n); | |
A = copy(D); | |
for idx = 2 : dims | |
A = kron(A, speye(n)) + kron(speye(size(A, 1)), D); | |
end | |
A | |
end | |
# (-1 2 -1) on the diagonal (actuall minus the laplacian) | |
second_order_central_diff{T}(::Type{T}, dim) = convert(SparseMatrixCSC{T, Int}, SymTridiagonal(fill(2 * one(T), dim), fill(-one(T), dim - 1))) | |
# For the rhs | |
f(x, y, z) = exp.(x .* y .* z) .* sin.(π .* x) .* sin.(π .* y) .* sin.(π .* z) |
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