Von Neumann Stability Analysis

The method above is known as Foward Time Centered Space (FTCS) . Unfortunately, contrary to the finite diffrence method used to solve Poisson and Laplace equation, the FTCS is an unstable method.

The fact that the solution depends on space and time introduces a criteria for the choice of the space grid and the time grid intervals. Assume that the coefficients of the finite difference equation are roughly constant. Then the eigenmode have the following form

$$T_{n,j} = \xi^nj e^{ n i k / Delta_x}$$ (6)

where $\xi(k)$ is some arbitry complex amplitude and k$k$ a constant. Substitution in the finite difference heat equation leads to this factor

$$\xi = 1 - \frac{2 \kappa \Delta_t}{ \Delta_x^2 }$$ (7)

Requiring that $\vert\xi\vert<1$ implies stability criterion (the nodes remain finite)

$$\frac{2 \kappa \Delta_t}{ \Delta_x^2 } < 1$$ (8)

This condition derives from the physical fact that the maximum allowed time interval $\Delta_t$ is, up to a numerical factor, the diffusion time across a cell of width $\Delta_x$.

Note that diffusion expands proportional to $\sqrt{t}$.