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using Pkg
Pkg.add("Plots")
using Plots
function del2_5(a::Matrix{Float64}, dx=1.0)
#=
Returns a finite-difference approximation of the laplacian of the array a,
with lattice spacing dx, using the five-point stencil:
0 1 0
1 -4 1
0 1 0
=#
del2 = zeros(size(a))
del2[2:end-1, 2:end-1] .= (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
a[3:end, 2:end-1] + a[1:end-2, 2:end-1] -
4.0 * a[2:end-1, 2:end-1]) ./ (dx^2)
return del2
end
function del2_9(a::Matrix{Float64}, dx=1.0)
#=
Returns a finite-difference approximation of the laplacian of the array a,
with lattice spacing dx, using the nine-point stencil:
1/6 2/3 1/6
2/3 -10/3 2/3
1/6 2/3 1/6
=#
del2 = zeros(size(a))
del2[2:end-1, 2:end-1] .= (4.0 * (a[2:end-1, 3:end] + a[2:end-1, 1:end-2] +
a[3:end, 2:end-1] + a[1:end-2, 2:end-1]) +
(a[1:end-2, 1:end-2] + a[1:end-2, 3:end] +
a[3:end, 1:end-2] + a[3:end, 3:end]) -
20.0 * a[2:end-1, 2:end-1]) ./ (6.0 * dx^2)
return del2
end
function invDel2_5(b::Matrix{Float64}, dx=1.0, N=10000)
#=
Relaxes over N steps to a discrete approximation of the inverse laplacian
of the source term b, each step is a weighted average over the four neighboring
points and the source. This is the Jacobi algorithm.
=#
invDel2 = zeros(size(b))
newInvDel2 = zeros(size(b))
for m in 1:N
newInvDel2[2:end-1, 2:end-1] .= 0.25 * (invDel2[2:end-1, 3:end] +
invDel2[2:end-1, 1:end-2] +
invDel2[3:end, 2:end-1] +
invDel2[1:end-2, 2:end-1] -
(dx^2) * b[2:end-1, 2:end-1])
invDel2 .= newInvDel2
end
diff = del2_5(invDel2, dx) - b
diffSq = diff .* diff
error = sqrt(sum(diffSq))
println("\nerror = ", error)
return invDel2
end
# Define variables
L = 10
dx = 1.0
# Parabola of revolution has constant Laplacian
phi = zeros((L, L))
for i = 1:L
x = (i + 0.5 - 0.5 * L) * dx
for j = 1:L
y = (j + 0.5 - 0.5 * L) * dx
phi[i, j] = x^2 + y^2
end
end
println(size(phi))
println(del2_5(phi, dx))
println("\n\n")
println(del2_9(phi, dx))
# Electrostatics example: uniformly charged cylinder (rho = 1) of radius R
L = 100
dx = 1.0
R = 20.0
R2 = R^2
# Charge distribution
rho = zeros((L, L))
for i = 1:L
x = (i + 0.5 - 0.5 * L) * dx
for j = 1:L
y = (j + 0.5 - 0.5 * L) * dx
r2 = x^2 + y^2
if r2 < R2
rho[i, j] = 1.0
else
rho[i, j] = 0.0
end
end
end
rhoPlot = plot(rho[Int(L/2), :])
# Exact (analytical) electric potential
phi = zeros((L, L))
for i = 1:L
x = (i + 0.5 - 0.5 * L) * dx
for j = 1:L
y = (j + 0.5 - 0.5 * L) * dx
r2 = x^2 + y^2
if r2 < R2
phi[i, j] = -π * r2
else
phi[i, j] = -π * (R2 + R2*log(r2/R2))
end
end
end
phiPlot = plot(phi[Int(L/2), :])
# Charge density obtained from exact potential
rho = -1.0/(4.0 * π) * del2_5(phi, dx)
rhoPlotLattice = plot(rho[Int(L/2), :])
phi = invDel2_5(-4.0 * π * rho, dx, 20000)
#phi = invDelSOR(-4.0 * π * rho, dx, 500)
phi .-= phi[Int(L/2), Int(L/2)]
phiPlotInvDel = plot(phi[Int(L/2), :])
contourf(phi)
#plot(rhoPlot, rhoPlotLattice)
#plot(phiPlot, phiPlotInvDel)
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