Hello,
The timestep can be calculated in cartesian coordinates using the formula
dt = (1/c)/sqrt( 1/dx^2 + 1/dy^2 + 1/dz^2)
But in cylindrical coordinates this gets more trickier. In lot of pares the timestep is set as
dt = (1/c)/sqrt( 1/dr^2 +1/(dr*df)^2 + 1/dz^2)
and in some others
dt = (1/c)/sqrt( 1/dr^2 +2/(dr*df)^2 + 1/dz^2)
taking into account that the smallest non-zero arc to be resolved by the solver is 0.5*dr. I am trying to figure out which one should I use because the first one produces larger timestep which is of course preferable. I know that the timestep can be also calculated as the solution of the numerical dispersion, which in cylindrical coordinates is a little bit tricky and problem dependent.
Thank you in advance
Dear Dimitrios, could you be more explicit about the equation you are solving? Looking at the stability constraints you mention, it seems that you are trying to treat a diffusion in explicit form. In this case, it would be worth in any case to consider an implicit time discretization.
Dear Roberto,
I am sorry I didn't refer the equations to be solved. I solve Maxwell equations explicitely using FDTD in cylindrical coordinate system. I know that implicit FDTD allows larger timestep but I refer to the explicit method, if there is an solid definition about the timestep
In my opinion, the easiest way to give a bound on the timestep is to write down the scheme explicitly in the matrix form U^{n+1}=BU^n+F^n, and then enforce monotonicity, i.e., require the discrete evolution operator B to have positive elements - an easy computation. This is how you derive the stability bound in the cartesian case.
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