As we know , some models in math and engineering lead to differential and partial differential equations with Nabla or laplacian operator that needs to be studied in x , y and z directions.So , the question is : what are the best and fastest numerical methods to solve them?. Another questions is : How to decompose them before solving ?.Because , we are facing with nabla or laplacian Operator in 3 directions.
Thank you so much in advance.All responses and comments will be acknowledged.
Dear. Kasmaei,
unfortunately your question is a little bit unspecific and therefore hard to answer completely. In general, what's relevant to determine the best numerical scheme depends on the type of the PDE. In particular whether it is elliptic, parabolic or hyperbolic. So, first find out what type your PDE is and then look for appropriate schemes.
As a rule of thumb, the Laplacian operator makes a PDE elliptic or parabolic. Good schemes for these PDE's are, e.g., Finite-Element methods. In turn, the Nabla operator makes a PDE hyperbolic and good schemes for such equations are, e.g., the Finite-Volume method or particle based or Lagrangian methods. If you have a mixed type (i.e., parabolic-hyperbolic), you need to find out which term is dominating by performing a scaling analysis.
These are, however, only rules-of-thumb and often the devil is in the detail. If you want to have better advice, you should specify the equation plus give some information about boundary and initial conditions, etc.
With respect to your second question, I am not sure I understand the question. Typically, you don't have to decompose anything. Numerical schemes like Finite-Elements are straight forward with respect to any number of spatial dimensions.
I would agree with this response. While it certainly depends on the form of your PDE, a finite-element method might be a good place to start. It is a pretty common method, so you can easily find resources to help you figure it out if you aren't familiar with how it works. Best of luck!
I agree with the previous answers. It depends on the nature of your PDE. If your new to the field, I recommand that you use any softaware that includes toolboxes already made to solve parttial differential equations (In matlab for example). If you want to develop your own codes, I do recommand high resolution central schemes for hyperbolic PDEs (http://www.cscamm.umd.edu/tadmor/pub/central-schemes/Kurganov-Tadmor.JCP-00I.pdf)
for parabolic PDEs, you can try some implicit finite difference schemes..
I think the best numerical methods is finite difference ,it 'is easy to implemnte .