Numerical Methods Help

[physmurf]

Numerical Methods Help!

I have been trying to understand the differences between Finite-Difference Time-Domain (FDTD), Finite Volume, and the Finite Elements methods of solving Maxwell's equations numerically. I have used the FDTD method for solving Maxwells Equations. I did this without knowing anything about the other two methods. So, now I need to know how FDTD is different from the other two.

Thanks

 

[clive]

In the finite elements approach you compute a solution of the differential equation on each element and then you "assemble" these individual solutions into a global one. So you solve your differential equation on each element.

In the finite difference approach you work always with global solutions. The differential operators are represented by matrices and the differential equation is reduced to a linear system of equations. Solving this system you obtain directly the solution of the differential equation.

Another difference consists of the fact that the finite element method is limited to a well defined class of differential equations (see Lax-Milgram theorem).

Hope it helps
clive

 

[physmurf]

That's a great help clive. You don't happen to know anything about the finite volume approach?

Thanks again

Kurt

 

[clive]

Here you have a good point:

http://mathworld.wolfram.com/FiniteVolumeMethod.html

clive

 

[LeBrad]

Finite volume methods are used when you have a conservation law to work with. Something like $$u_t + f_x = 0$$ where f(u) is a flux. Rather than just evaluating u at grid points as you do in finite differences, you need to average u over little cells to get the value for a point. You have to make sure what goes out of cell i is going into cell i+1. Finite volume methods can be useful if you're trying to handle a shock without screwing everything up.

 

[SpaceTiger]

Finite volume methods can be useful if you're trying to handle a shock without screwing everything up.

What should you use if you do want to screw everything up?

 

[HallsofIvy]

Ah, now I'm an expert on that!

 

[LeBrad]

What should you use if you do want to screw everything up?

Finite differences would be a good start. If you apply a finite difference method and a finite volume method to the same problem which contains a shock, the finite differences method will probably give you garbage and the finite volume method will probably give you yummy stuff.

A non-conservative method is likely to give you a solution which propagates the shock at the wrong speed. This results in incorrect/inconsistent fluxes across the domain. If you're using a finite volume scheme derived from a conservation equation this will never happen, because an incorrect flux would mean your thing is not being conserved, and these methods don't allow that. Of course you could still get the wrong answer for other reasons.

In fact, a finite volume scheme can give many solutions for a problem with a shock, and mathematically they may all work. But only one solution will make physical sense, so you need to use an entropy satisfying scheme to automatically pick the right solution.