I’m solving the convection-diffusion equation using the finite element method, and I have a question about the time step (dt) and spatial step (dx) limits.
From what I’ve read, I understand that the CFL condition and the Fourier number set the maximum limits for dt and dx to ensure stability and accuracy. However, I couldn’t find any information about the minimum limits for dt and dx.
In regards to the CFL number, it should be pointed out that this is only a stability requirement for explicit time integrators. For linear problems, most implicit algorithms are unconditionally stable and have no restrictions on the timestep (assuming the eigenvalues of your spatial discretization live in the left-half of the complex plane).
In terms of minimal values, no there is no theoretical limit. If a scheme is consistent (and stable), then the discrete approximation will approach the true solution in the limits to zero of dt and dx (assuming the CFL condition is met at all times, if an explicit scheme is used).
Practically, too small values can make the linear algebra solve challenging. Not only do small values mean many time solves and huge system matrices, the solve step in each timestep also becomes more difficult due to matrix conditioning, and numerical round-off.
The main practical tip is simply, ‘‘don’t use a finer discretization (in space and time) than is needed to solve your problem to the accuracy level that you need’’. That requires a convergence study, where you see if your ‘quantities of interests’ no longer change with further refinement.
Thank you very much for your reply. Your explanation was very clear and helpful. I’d like to ask for further guidance regarding my specific system parameters, which are as follows:
flow velocity: 0.25 m/s
diffusion coefficient: 1.12e-8
concentration field: values range from 1 to 100
time step (dt):1.00e-4
spatial step (dx): 0.01
My questions are:
Based on these parameters, do you think the current values of dt and dx are appropriate?
Are they accurate enough to approximate the real solution, or could they be too small and prone to numerical errors (e.g., round-off issues)?
If possible, could you suggest any adjustments to improve accuracy or efficiency in solving this problem?
Again, thank you very much for your time and valuable insights!
I’d encourage you to study arbitrary order methods to get a better understanding how to answer this question yourself. E.g., the books of R. J. LeVeque and Runge Kutta methods.
Although we can study your problem for you and give you a rough idea, only you know what will be satisfactory for your needs.
Every modeller should go through a similar process of convergence studies and theoretical analysis to build confidence in their computed result.
