Consider the PDE
$$\frac{\partial f}{\partial t} = \alpha\frac{\partial^2 f}{\partial x^2}$$
- Show how to nondimensionalize this PDE.
- What is the characteristic timescale $\tau$ for the PDE?
- We form the ratio $d=\Delta t/\tau$, which is the nondimensional time step. What is the max value of $d$ for stability?
- (Intuitively, for stability, you should not take a step that is larger than some factor times the intrinsic physical timescale. Even if stable, e.g., for an implicit method, you should not take too large a time step in order to get accurate results.)
- Without doing all the math, outline the Von Neumann stability analysis. What are the key steps? Aim for comprehension of what is happening, rather than just following the math from one step to the next.
- Suppose the PDE is nonlinear, like, $\alpha=\alpha(f)$. How could we analyse the stability?
- In practice, how would you set the time step if $\alpha$ varies on the domain?