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Maxwell solver using Yee scheme

$L_x,L_y$ domain dimensions and M,N are integers.

\[\omega = \sqrt{(\frac{M\pi}{L_x})^2+(\frac{N\pi}{L_y})^2}\]

\[B_z(x,y,t) = - \cos(M \pi \frac{x}{L_x}) \cos(N \pi \frac{y}{L_y}) \cos(\omega t)\]

\[E_x(x,y,t) = \frac{c^2 N \pi }{\omega Ly} cos(M \pi \frac{x}{L_x}) \sin(N \pi \frac{y}{L_y}) \sin(\omega t)\]

\[E_y(x,y,t) = - \frac{c^2 M \pi }{\omega Lx} \sin (M \pi \frac{x}{L_x}) \cos (N \pi \frac{y}{L_y}) \sin(\omega t)\]

using ParticleInCell

dimx, dimy = 1, 1
nx, ny = 64, 64
md, nd = 2, 2
dt = 0.001
nstep = 1 ÷ dt
mesh = TwoDGrid( dimx, nx, dimy, ny )
maxwell = FDTD( mesh )
omega = sqrt((md*pi/dimx)^2+(nd*pi/dimy)^2)

x = 0.5 .* (mesh.x[1:end-1] .+ mesh.x[2:end])
y = 0.5 .* (mesh.y[1:end-1] .+ mesh.y[2:end]) |> transpose

maxwell.bz .= - cos.(md*pi*x) .* cos.(nd*pi*y) .* cos(omega*(-0.5*dt))

surface(maxwell.bz, aspect_ratio=:equal, zlims=(-1,1))
Example block output
  • Ex and Ey are set at t = 0.0
  • Bz is set at t = -dt/2
function run(mesh, maxwell, nstep)

    x = 0.5 .* (mesh.x[1:end-1] .+ mesh.x[2:end])
    y = 0.5 .* (mesh.y[1:end-1] .+ mesh.y[2:end]) |> transpose

    maxwell.bz .= - cos.(md*pi*x) .* cos.(nd*pi*y) .* cos(omega*(-0.5*dt))

    nx, ny = mesh.nx, mesh.ny
    jx = zeros(nx+1, ny+1)
    jy = zeros(nx+1, ny+1)

    @gif for istep = 1:nstep # Loop over time

        faraday!(maxwell, mesh, dt)

        ampere_maxwell!(maxwell, mesh, jx, jy, dt)

        surface(maxwell.bz, aspect_ratio=:equal, zlims=(-1,1), clim=(-1,1))

    end every (nstep ÷ 100)


end

run(mesh, maxwell, 2000)
Example block output