Comparative numerical laboratory
PDE Schemes: Heat Equation Stability and Animation — Chapter 12
Animate the explicit heat scheme and compare stable/unstable choices of the Courant number.
\[
u_i^{n+1}=u_i^n+\mu(u_{i-1}^n-2u_i^n+u_{i+1}^n), \mu=\frac{\kappa\Delta t}{\Delta x^2}.
\]
Methods compared
| Method | Convenience | Inconvenience / warning |
|---|---|---|
| Explicit heat scheme | Simple and easy to animate. | Stable only under a step-size restriction. |
| Implicit viewpoint | Stable for larger time steps. | Requires solving linear systems. |
| Crank--Nicolson viewpoint | Second order in time for smooth solutions. | Can oscillate for nonsmooth data if not damped. |
Illustrative plot / animation
Numerical diagnostics
What the student must compare
- CFL condition
- mode amplification
- time animation
- mesh/time-step effect