Can Iterative Methods Optimize Non-Linear Finite Differences?

[Pete99]

Hi all,

I am trying to solve a non linear differential equation iteratively using finite differences.

At every iteration I basically have to solve (sorry, for some reason I cannot use the preview function when I write latex, so I'll write in plain text):

Delta_x = (A-J)\b



Where Delta_x is the update I calculate in each iteration, A is a sparse matrix with 5 non zero diagonals and J is diagonal. b and J depend on x, the parameter that I am looking for.

The diagonal matrix J, has only about 10% of its diagonal elements non zero (at fixed positions) and change at every iteration, and A does not change during the iterations.

Right now I calculate the update using the backslash operator in Matlab. My question is if there is any way I can take advantage of the fact that only a small fraction of J changes at every iteration in order to save time.

Thanks,

 

[jvicens]

I would suggest considering using an iterative method such as the Jacobi or Gauss-Seidel method to solve your non-linear differential equation. These methods would allow you to take advantage of the fact that only a small fraction of J changes at each iteration, as they only update one element at a time. This can potentially save time and improve the efficiency of your solution.

Additionally, you may want to look into using a sparse matrix solver instead of the backslash operator in Matlab. This can also help save time and improve the accuracy of your solution, especially for large sparse matrices like the one you mentioned.

Finally, I would recommend considering other numerical methods such as finite element or finite volume methods, which may be more suitable for solving non-linear differential equations. These methods can handle complex geometries and boundary conditions, and can also be more efficient for solving non-linear problems.

I hope this helps and good luck with your research!