Richardson Extrapolation to check convergence

In summary, the conversation discusses using Richardson Extrapolation to check space and time convergence without the need for further simulations using smaller time steps or a finer mesh. Richardson invented this method for numerical integration and it can be found in the "Numerical Recipes" book. The method involves using two differently sized course meshes and then extrapolating to find the solution for a finer mesh. It can save a lot of time in simulations.
  • #1
mina1363
14
0
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
 
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  • #2
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.
 
  • #3
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
 
  • #4
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?
 
  • #5
mina1363 said:
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.
 
  • #6
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.
 

1. What is Richardson Extrapolation?

Richardson Extrapolation is a numerical method used to improve the accuracy of a sequence of approximations to a desired value. It is commonly used in mathematics and engineering to check the convergence of a series or to estimate the value of a function at a point.

2. How does Richardson Extrapolation help with convergence?

Richardson Extrapolation works by taking two or more approximations and combining them in a way that eliminates the largest error term, resulting in a more accurate estimate. This process can be repeated multiple times to further improve the accuracy of the estimate.

3. What is the formula for Richardson Extrapolation?

The formula for Richardson Extrapolation is:
f(x) = [2nfh(x) - f2h(x)] / (2n - 1)
where n is the number of iterations, fh(x) and f2h(x) are the two approximations with step sizes h and 2h, respectively.

4. How do you know when to stop using Richardson Extrapolation?

There are a few ways to determine when to stop using Richardson Extrapolation. One method is to set a tolerance level and continue the process until the difference between two consecutive iterations is less than the tolerance. Another way is to monitor the change in the estimate with each iteration and stop when it becomes negligible.

5. Can Richardson Extrapolation be used for any type of series or function?

Yes, Richardson Extrapolation can be used for any type of series or function as long as the approximations used in the formula are valid. However, it is most commonly used for power series and integrals.

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