# Stability analysis of coupled ordinary differential equations

Given a forward-in-time approximation I have the coupled equations: $$\frac{T^{(n+1)} - T^{(n)}}{\Delta t} = x T^{(n)} - y h^{(n)} \\ \frac{h^{(n+1)} - h^{(n)}}{\Delta t} = -z h^{(n)} - \alpha T^{(n)}$$ where $x, y, z$ and $\alpha$ are constants, I see from my simulation that the solution is damping. But how can I use Von Neumann analysis to find the amplication factor $A$?

• can someone explain von neumann stability analysis for system of equation – Farhat Asim Dec 8 '16 at 14:27

Rewritten this way, you will find that the matrix $A$ becomes $$A=I+\Delta tJ,$$ where $I$ is the identity matrix and $J$ is the jacobian of the RHS of your original system. $A$ can be thought of as the amplification matrix, whose eigenvalues determine the stability of the numerical system. To ensure absolute stability, you need to find a region in the complex vector space such that the spectral radius $\rho(A)<1$ (i.e. the dominant eigenvalue is less than one).