Computational Physics III Diffusion Equation

9. Diffusive Partial Differential Equations

Random walks
- one-dimensional symmetric random walk (demo)
- mean and variance
- probability distribution (many walkers) (demo)
The diffusion equation
- difference equation for a random walk
- equivalent partial differential equation -- diffusion equation
- the fundamental solution of the diffusion equation
- aside: connection with the Schrodinger equation
Discretization and differencing of the diffusion equation.
Exercise 9.1: Solution by explicit differencing. The goal is to reproduce a portion of the fundamental solution.
Take the diffusion coefficient to be D = 1, and work on a uniform grid of 201 points running from x = -10 to x = 10. Start with the fundamental solution at t = 1 and integrate the system forward in time to t = 10 in steps of size dt. Plot your solution at t = 10 for dt = 0.005, 0.01, 0.05, 0.1, and compare it to the fundamental solution at t = 10.
You can use as a starting point the solution to Exercise 6.1. Your task is to rewrite the FTCS integrator to implement the diffusion equation. [Solution]
Stability of the integration scheme -- von Neumann analysis.
Problems with the explicit scheme -- scaling.
Stability and implicit differencing -- the fully implicit scheme.
Solving a tridiagonal matrix equation -- tridiag.py.
Exercise 9.2: Repeat exercise 9.1, but using the fully implicit differencing scheme. [Set-up code] [Solution]
Splitting the difference -- the Crank-Nicolson scheme.
Exercise 9.3: Solution by semi-implicit differencing. Repeat exercise 9.1, but using the Crank-Nicolson scheme with a time step of 0.1.
Exercise 9.4: The temperature T of a uniform medium is governed by the diffusion equation with diffusion coefficient D. The medium extends from x = 0 to x = L. Initially the medium is at temperature T = 0, and then the temperature of the left edge (x = 0) is raised to T[0] = 1 and held at that value. Adopt boundary conditions so that T[0] = 1 and T'[J-1] = 0. Use the Crank-Nicolson scheme with J = 1001 points to determine the time taken for the right edge (x = L) to reach a temperature of T[J-1] = 0.5 for (i) D = 1, L = 10, (ii) D = 4, L = 10, and (iii) D = 1, L = 5. Do these times agree with our earlier estimates of the time scale for diffusion across the grid?