Richardson Extrapolation to check convergence

[mina1363]

Hi,

I need to know how one can check space and time convergence using Richardson Extrapolation. Does anyone know any good references. I have a slight idea... the thing I am wondering about is how using this method can eliminate the need for further simulations using smaller time steps or a finer mesh.

Thanks

 

[AlephZero]

Yes, that's what Richardson invented it for!

A fairly simple application of it to numerical integration is called the Romberg method. There is a good description in the "Numerical Recipes" book. That should get you started understanding how it works.

 

[Topher925]

What is it exactly you don't get? The method is pretty strait forward. Take the results from two differently sized (h, h/2) course meshes, plug the results into the general formula, and then that's it. If the deviation of your results are within the limits of error you need, then you're good.

Any good text on numerical methods, FEA, or CFD will go into the details of it.

http://en.wikipedia.org/wiki/Richardson_extrapolation

 

[mina1363]

Many thanks. I understand the method.

I am confused because I was told to use Richardson extrapolation to eliminate the need of using a refined mesh or smaller time steps. However From what I understand and also what you just told me the method is based on using the step/mesh size so I would still need to refine the mesh or use a smaller time step to check the space and time convergences respectively...so am I correct in saying that what I was told is wrong?

 

[Topher925]

Many thanks. I understand the method.

I am confused because I was told to use Richardson extrapolation to eliminate the need of using a refined mesh or smaller time steps. However From what I understand and also what you just told me the method is based on using the step/mesh size so I would still need to refine the mesh or use a smaller time step to check the space and time convergences respectively...so am I correct in saying that what I was told is wrong?

No, what you were told was correct. For what ever numerical model you have, you will want to achieve mesh independence. In other words, the answer won't be dependent upon the size of your mesh. For example, if you have a mesh with spacing h, your results won't change beyond some relative error (maybe 1%).

Sometimes in order to achieve mesh independence, a mesh or step size greater than the computer can handle is required. So, to achieve a solution with mesh independence you extrapolate using two course mesh sizes that the computer can handle.

For example, you need a step size of "h" to achieve mesh independence but your computer can only handle a step size of 2h or greater. Using Richardson extrapolation you can estimate the solution for h by extrapolation using the results for step sizes of 2h and 4h. In the end, you end up with a solution for h by finding the solutions for 2h and 4h which are less computationally intensive.

 

[mina1363]

Many thanks for making me realize this.

I was too focused on the fact that I have to remesh and run the simulation again and not thinking that the point was about a finer mesh and not a coarser one... so I'm just going to run the simulation with a coarser mesh and use the extrapolation which would save a lot of time in my case.

Thanks again.