Von Neumann Stability Analysis
Neumann’s analysis is a method that assesses the stability of a numerical scheme. John Von Neumann is a Hungarian-American mathematician who had invaluable contributions to the field of numerical computation. He also participated in the nuclear Manhattan project.
Briefly, the method consists of using the finite Fourier series decomposition of the numerical solution to determine whether it will eventually blow up, hence becomes unstable or will it remain stable. I will show you a detailed example of how the method works using a 1D linear convection equation with an explicit scheme that is forward difference in time and central difference in space.
Let’s start with the numerical discretization of the equation:
Let’s replace with
Let’s divide with
Using common knowledge:
Let’s define a variable F:
The condition for a stable numerical scheme is:
< 1 where is the absolute value of so after calculation we have:
which will always be >1, hence the scheme is unconditionally unstable.