Are the usual methods like central difference which is widely used to model a diffusion dominant situation not likely to be fit for modeling convection dominated scenarios, as it produces spurious oscillations? I observed that increasing the number of points in convection calculations reduces dispersion error (which is due to fact of increased accuracy ), but it also leads to interpolating the boundary conditions. How can I choose the best suitable method for given convection dominated 2D problems?
These are known stability issues. What the dispersion error means isn't clear. I'd suggest using a pseudo-spectral method, i.e. expansion in some basis of functions, that satisfy the boundary conditions and use fast transforms.
I expect that you are using control volume or a similar finite difference approximation? Then the central differences can work well only if the advection is not strong, i.e. if your Peclet number = abs(density*velocity*length/difussion coefficient) is larger than 2, the central difference fails. Instead you should use up-wind scheme (which is 1st order scheme in contrast with central difference which 2nd order scheme). But the up-wind introduces false diffusion, whereas the central difference not, but as I said above, the central difference fails for large convection dominated flow. There are other numerical schemes like QUICK and others. Check the books by Patankar, or Ferziger & Peric or Malalasekera.
I have been using both central difference and spectral method for simulation of flow in porous media. Spectral method is very accurate and has much less numerical instability or false diffusion. However you need to either apply periodic boundary conditions or multiple your domain size to mirror the domain for achieving periodicity. Some also use separation of variables which is helpful to extract the non-periodic part of the solution and solve it analytically.
For a specific problem I found finite difference method more useful because in spectral method you cannot use non-uniform grid or the transformation becomes very complex and loses the advantage of accuracy.
All said, I would recommend spectral method if applicable!
No, periodic boundary conditions aren't necessary to apply (pseudo)-spectral methods. Once more, they involve expanding in a set of functions, that, themselves, satisfy the boundary conditions and for which fast transforms act efficiently.
The notion of a grid doesn't make sense in this context: the relevant parameter is the resolution in the functional space. To obtain values at particular spatial or temporal resolution, one uses interpolation. To do this, in fact, Chebyshev polynomials do satisfy ``optimality'' conditions that can be quantified.
Use 2nd order upwind , or Van Leer Schemes.
Incidentally central differences will produce oscillations for Peclet no,s outside the range -2 to 2.This is the reason that when you refine the grid you get solution,simply because you lower the Peclet no and you bring it within the above limits.
Depending on your goal (which is not stated), you might want to try [gerris](http://gfs.sourceforge.net/wiki/index.php/Main_Page), a flow solver that is powerful and well-tested.
You didn't state what your approximate Peclet number range was - as other post-ers have noted, the advection-diffusion problem with centered differences is strongly constrained in that the diffusion must 'dominate' advection or the solution is no longer consistent with the guiding PDE.
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