Hello, I want to know how I can check the stability of nonlinear system of equation. For example, if I have the following where everything is partial derivative:

du_dt + f(u) du_dx = 0

then, how can I use Neumann stability analysis to analyize a particular discretization of this equation. Any help or references to textbook type information would be most kind.

Linearity of the equation is a general requirement for the application of the von neumann stability analysis. But in this case your equation is non-linear so , locally linearize the non-linear equation and subsequently apply the von Neumann stability analysis.

I hope this will help.

rvndr

O.k. Thank you for the help. How would you suggest I locally linearise it? Also, how can I show that if the equation is locally lineraised, that it reflects the non-linear version well? In other words, how do I see if linearising it destroys any relation to the orgiinal non linear equation.

when dealing with linear equations you often only need to examine the behaviour of a single component, as all other will behanve similarly.

linearisation of your equation depends on the function of u, i.e. 1D linearlised Burgers equation: du_dt + c du_dx = 0 where c is constant and positive. this is a hyperbolic equation and leads to the CFL condition.

if you assume an explicit scheme in your original equation: du_dt + f(u) du_dx = 0 linearisation is trivial, by assuming old values. for an implicit scheme a linearisation method is needed, or alternatively an iteration within a time step can be used. look at the work by beam and warming where it is suggested to expand the flux function using a taylor series.

hope this helps