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Linear Stability Analysis Routine. Normal Modes in NDSolve

I want to make the linear stability analyis of a nonlinear system of PDEs. So for example then I want to plot what happens to the amplitudes of the modes. To do that I need to solve my equations with ...
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How to plot the Euler's formula in 3D?

In Von Neumann's stability analysis of Finite Difference Equations, the Euler's formula is used to describe a perturbation. Now the finite difference equation is stable if this perturbation does not ...
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How to demonstrate lack of stability with advection equation

I am trying to that using a coarse grid with an explicit method for, say, the advection equation leads to an unstable solution. The trouble is Mathematica avoids unstable solutions with good ...