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.




Methodology

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:

U_{i}^{n+1}= U_{i}^{n}-\frac{K}{2}\left (U_{i+1}^{n}-U_{i-1}^{n} \right )

Step 1

Let’s replace  U_{i+b}^{n+a} with W^{n+a}.e^{I(i+b).\Theta }

W^{n+1}.e^{I.i.\Theta }=W^{n}.e^{I.i.\Theta }-\frac{K}{2}.\left (W^{n}.e^{I.(i+1).\Theta }-(W^{n}.e^{I.(i-1).\Theta } \right ))

Step 2

Let’s divide with e^{I.i.\Theta }\Rightarrow W^{n+1}=W^{n}.(1-\frac{K}{2}.(e^{I.\Theta }-e^{-I.\Theta }))

Using common knowledge:

e^{a+b.I}=e^{a}.\left ( cosb+I.sinb \right ) (1)

Let’s define a variable F:

F=\frac{W^{n+1}}{W^{n}}=1-\frac{K}{2}.\left ( 2.Isin\Theta \right )=1-K.I.sin\Theta

The condition for a stable numerical scheme is:

F.F^{*} < 1  where F^{*} is the absolute value of F so after calculation we have:
1+K^{2}sin^{2}\Theta which will always be >1, hence the scheme is unconditionally unstable.

Unstable numerical solution
Unstable numerical solution




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