# Stability analysis for numerical schemes of systems of PDEs

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• hunt_mat
hunt_mat
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TL;DR Summary
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.

In principle, you can use the same technique: define the discrete fourier transforms of both variables, $$\begin{split} \hat\nu(\zeta, t) &= \sum_{n=-\infty}^\infty \nu_n(t)e^{in\zeta} \\ \hat u(\zeta, t) &= \sum_{n=-\infty}^\infty u_n(t)e^{in\zeta}\end{split}$$ Then taking the DFT of your PDEs you have the system $$\begin{split} \frac{\partial \hat\nu}{\partial t} &= N(\hat u, \zeta) \\ \frac{\partial \hat u}{\partial t} &= U(\hat u, \hat \nu, \zeta) \end{split}$$ 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 $\zeta \in [0, 2\pi]$.

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