Animation of the diffusion equation

Here is one approach (set the inner radius to 0 to use a circle instead of a ring).
Animation of the time evolution of the two-dimensional diffusion equation.
diffusion2danimation
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation

# plate size, mm
w = h = 10.0
# intervals in x-, y- directions, mm
dx = dy = 0.1
# Thermal diffusivity of steel, mm2.s-1
D = 4.0

Tcool, Thot = 300, 700

nx, ny = int(w / dx), int(h / dy)

dx2, dy2 = dx * dx, dy * dy
dt = dx2 * dy2 / (2 * D * (dx2 + dy2))

u0 = Tcool * np.ones((nx, ny))
u = u0.copy()

# Initial conditions - ring of inner radius r, width dr centred at (cx,cy) (mm)
r, dr, cx, cy = 2, 1, 5, 5
ri2, ro2 = r**2, (r + dr) ** 2
for i in range(nx):
    for j in range(ny):
        p2 = (i * dx - cx) ** 2 + (j * dy - cy) ** 2
        if ri2 < p2 < ro2:
            u0[i, j] = Thot


def do_timestep(u0, u):
    # Propagate with forward-difference in time, central-difference in space
    u[1:-1, 1:-1] = u0[1:-1, 1:-1] + D * dt * (
        (u0[2:, 1:-1] - 2 * u0[1:-1, 1:-1] + u0[:-2, 1:-1]) / dx2
        + (u0[1:-1, 2:] - 2 * u0[1:-1, 1:-1] + u0[1:-1, :-2]) / dy2
    )

    u0 = u.copy()
    return u0, u


# Number of timesteps
nsteps = 500
fig = plt.figure()
ax = fig.add_subplot()
# The initial image: interpolate for smoothness and use a perceptually uniform
# sequential colourmap.
im = ax.imshow(
    u0,
    cmap=plt.get_cmap("magma"),
    vmin=Tcool,
    vmax=Thot,
    interpolation="bicubic",
)
ax.set_axis_off()
ax.set_title("0.0 ms")
# Add in the colourbar on the right.
fig.subplots_adjust(right=0.85)
cbar_ax = fig.add_axes([0.9, 0.15, 0.03, 0.7])
cbar_ax.set_xlabel("$T$ / K", labelpad=20)
fig.colorbar(im, cax=cbar_ax)


def animate(i):
    """Advance the simulation and animation by one time step."""
    global u0, u
    u0, u = do_timestep(u0, u)
    ax.set_title(f"{i * dt * 1000:.1f} ms")
    im.set_data(u.copy())


interval = 10
ani = animation.FuncAnimation(
    fig, animate, frames=nsteps, repeat=False, interval=interval
)
plt.show()
P7.7.2Animation of the diffusion equation
