Graduate Stability analysis for numerical schemes of systems of PDEs

[hunt_mat]

TL;DR

Everyone knows how to do this type of analysis for a single equation, but what about systems?

I want to solve the following system of PDEs:
\frac{\partial\nu}{\partial t}=\frac{\partial u}{\partial h}
\frac{\partial u}{\partial t}=\frac{\partial}{\partial h}\left(f(\nu)\frac{\partial u}{\partial h}\right)

I know the usual Fourier analysis that are applied to the stencil for single equations that lead to conditions of the variables but I want to know how it's done for a system. I want to do the weighted-average method and I want to work out the value to give the largest dt.

 

[pasmith]

In principle, you can use the same technique: define the discrete fourier transforms of both variables, \begin{split}<br /> \hat\nu(\zeta, t) &amp;= \sum_{n=-\infty}^\infty \nu_n(t)e^{in\zeta} \\<br /> \hat u(\zeta, t) &amp;= \sum_{n=-\infty}^\infty u_n(t)e^{in\zeta}\end{split} Then taking the DFT of your PDEs you have the system <br /> \begin{split}<br /> \frac{\partial \hat\nu}{\partial t} &amp;= N(\hat u, \zeta) \\<br /> \frac{\partial \hat u}{\partial t} &amp;= U(\hat u, \hat \nu, \zeta) \end{split} to which you can apply the stability analysis, treating \zeta \in [0, 2\pi] as constant. The result is that both of the eigenvalues of the jacobian of this system must be in the stable region of the time integration for all values of \zeta \in [0, 2\pi].