My short answer would be that it depends on the dimension and accuracy required for the problem you are interested in. In atmospheric modelling for instance, the choice of the advection scheme is largely a compromise between the desired accuracy of the numerical method, its numerical stability and the computational cost required to run your model. Other important issues such as preservation of the monotonic character of tracers for example also come into play.

My short answer would be that it depends on the dimension and accuracy required for the problem you are interested in. In atmospheric modelling for instance, the choice of the advection scheme is largely a compromise between the desired accuracy of the numerical method, its numerical stability and the computational cost required to run your model. Other important issues such as preservation of the monotonic character of tracers for example also come into play.

I am doing modelling of pulse tube cryocooler which works by use of oscillating helium flow (cyclic flow - to and fro from the system). The system is having different components with high temperature gradients. Which scheme is suitable for my coding? I am currently using First order upwind.

I am absolutely not familiar with the problem you are studying. However, given that you are dealing with large spatial gradients, I would be tempted to experiment with more accurate higher-order schemes.

In my opinion the proper choice of upwind schemes has to be related also to the amount of numerical dissipation you are aware to introduce in your simulation.

In numerical simulation of turbulent flows where large gradients in velocity field are generated by the three dimensional vortex stretching, a first order upwind scheme for advection would infer the correct energy transfer mechanism which has to be inviscid.

I'm not familiar too with the problem you are dealing with, but if your system but if your velocity field has a lot frequency components I would prefer higher order upwind schemes to discretize the advection term.

Upwinding usually comes with compressible flows, where there are bias propagation of information. I would try to see which upwinding scheme follows the physics of the flow more. Anotherwords, the more physics put into the upwinding scheme, it makes it more realistic. I have found Roe's FVM scheme suitable for most applications, especially when shock waves are present. Upwind scheme has to be smart enough to adjust itself for different situations.

Convection by nature has an "upwind" element.For example if there is a smell downwind of me I will smell it only if the wind blows against me! Therefore any scheme must have an"upwind"element (this is why central differences fail for large Peclet numbers, outside the range -2 to 2). First order upwind has some false diffusion and require finer grids for accuracy.Quick presents wingles.

Best schemes are Flow-Oriented schemes (see for example D.P.Karadimou and N.C.Markatos,A novel flow -oriented discretisation scheme .....,Atmospheric Environment,61,p327-339,2012)

For details and critical evaluation please see:

M.K.Patel et al "An evaluation of eleven disctetisation schemes......" Int.J.Heat Mass Transfer,vol 30,9,p1907-1925,1987.Also

"Methods for reducing False Diffusion......" Applied Mathematical Modelling,vol9,4,p302-306,1985 and 'A critical evaluation of seven discretisation schemes.......' Int.J.Numer.Methods inFluids,vol5,p225-244,1985.

For general use I prefer 2nd order Upwind or SUCCA (see refs above)

Rather easily.

The flux entering a cell face is ρuφ ,φ being any dependent variable.This φ must be the value at the previous point in the direction of flow to the cell.