Diffusion in multi-dimensions

Diffusion in multi-dimensions#

Sometimes it can be useful to mix explicit and implicit methods. In 1D, the Crank-Nicholson method is

T_{i}^{n + 1} - T_{i}^{n} = α [\frac{1}{2} (T_{i - 1}^{n + 1} - 2 T_{i}^{n + 1} + T_{i + 1}^{n + 1}) + \frac{1}{2} (T_{i - 1}^{n} - 2 T_{i}^{n} + T_{i + 1}^{n})]

where on the right-hand side we take the average of the explicit and implicit updates. This method is described as semi-implicit. Also stable for any $Δ t$ , the advantage of Crank-Nicholson over the fully-implicit method is that it is second order in time instead of first order.

The alternating-direction implicit scheme is a way to solve the 2D diffusion equation

\frac{\partial T}{\partial t} = \frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} .

The idea is to split the update into two half timesteps:

T_{i, j}^{n + 1 / 2} = T_{i, j}^{n} + \frac{α}{2} (T_{i - 1, j}^{n + 1 / 2} - 2 T_{i, j}^{n + 1 / 2} + T_{i + 1, j}^{n + 1 / 2} + T_{i, j - 1}^{n} - 2 T_{i, j}^{n} + T_{i, j + 1}^{n})

T_{i, j}^{n + 1} = T_{i, j}^{n + 1 / 2} + \frac{α}{2} (T_{i - 1, j}^{n + 1 / 2} - 2 T_{i, j}^{n + 1 / 2} + T_{i + 1, j}^{n + 1 / 2} + T_{i, j - 1}^{n + 1} - 2 T_{i, j}^{n + 1} + T_{i, j + 1}^{n + 1})

In the first step we treat the $x$ -direction update implicitly and the $y$ -direction explicitly, and then alternate for the second step.

Exercise: 2D diffusion

Implement this scheme to solve for the time evolution of the temperature profile in a 2D square box that is at $T = 0$ initially. One side of the box is held at $T = 1$ for $t > 0$ , the other sides are kept at $T = 0$ .

[Solution]