I am reviewing stability of finite difference methods. I am using the von Neumann method to study a linear PDE and an explicit finite difference scheme. The goal is to develop an expression for an amplification factor $G$.

I've read that if $G$ is a complex function it can be plotted in the complex plane to more easily determine where the solution of the finite difference equation will be stable. For example, consider if $G$ was

$$ G = (1-2d) + 2d\cos\theta $$

where $d$ is determined from my grid and time step sizes.

What is the template or formulas used to re-write this in terms of a circle (or some other shape, whatever it is) in the complex plane? Once I know that, I can plot it and hopefully tell where the stable region is.

My reference indicates this example of $G$ is an oscillation centered at $(1-2d,0i)$ with an amplitude of $2d$. It isn't clear to me how that was determined or how you could use that to plot it.


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