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I have a typical 1D advection problem where a cold fluid flows over a flat plate. I did an energy balance to include conduction, convection and friction loss and I got the PDE's for the fluid and the solid. I used finite differences to solve the system as T(x, t) for both fluid and solid. After simplification, I have this kind of matrix in explicit scheme.

(see attached matrix.png)

The problem is that I want to optimise the number of nodes (time step) in my simulation to decrease time of calculation and ensure stability and convergence. I read about CFL criteria but it doesn't seem to always work in my case even if CFL < 1. Here (advection.png) is an image of the temperature distribution at a specific node for 2 different cases. By trial and error, I determined that the lowest minimum of nodes would be around 20 to get the full phenomen. However, is there a analytical way to figure out this value ?

Thank you,

Steven

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# Advection equation stability for explicit scheme

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