# Advection equation stability for explicit scheme

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1. Aug 25, 2015

### pomekrank

Hello,

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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2. Aug 27, 2015

### Geofleur

The only way I know to do this is to perform a grid resolution study. You would start with a coarse grid, run the simulation, and keep refining the grid until the changes in the computed solution fall below a certain tolerance. You might consider automating this procedure. That is known as dynamic grid adaptation.

3. Sep 2, 2015

### the_wolfman

Von Neumann stability analysis is the standard technique for accessing the stability of a finite difference scheme.

However, some stable finite difference schemes will have spurious oscillating modes if the time step is too large. An example of this is the Crank-Nicolson method applied to the heat equation. While these mode eventually decay to zero, they can still negatively impact a calculation. Your plot of the temperature is indicative of such behavior.