Week 9 - FVM Literature Review

Finite Volume Method is a discretization method for the approxaimation of a single or a system of partial differential equations which conserve one or more quantities in a given system. Unknown field variables such as temperature, pressure, concentration, density can be determined with the help fof these conservation equations through the process of finite volume method.
For a given system, a mesh is contructed and the elements of this mesh are called control volumes. The integration of the PDE or conservation equation over each control volume provides a balance equation which is then discretized tosolve for a specific unknown variable. A system of equations can be generated for the control volumes and that would solve for the unknown variable over the entire system.

Finite Volume Method	Finite Difference Method
1. It is an integral scheme.	1. It is a differential scheme.
2. Has lower error values.	2. Has slightly higher error values.
3. Is slightly slower.	3. Is slightly faster.
4. Is a superior method when the mesh is irregular.	4. Is inferior when the mesh is irregular.

The approximation of the surface and volume integrals may require values of the unknown variable at locations other than the grid points as defined by the mesh structure. To determine the values at those locations, interpolation schemes are required. Some schemes are Upwing Interpolation Scheme, Quadratic Upwind Differencing Scheme, Total Variation Diminishing Scheme, etc. For example in the heat conduction equation, thermal diffusivity is a function of temperature. As the conservation equation is integrated over a control volume, one would need the value of thermal diffusivity at a point. For that, they would need to know the temperature at that point, for which the interpolation scheme is necessary.

Flux Limiters are used to tune high order and low order schemes such that the resulting scheme gives a higher order of accuracy in the smooth region of flow and maintains first order accuracy in the vicinity of shocks and discintinuities. This is required since in convective flow, low-order schemes are usually stable but highly dissipative around points of discontinuity or shocksm, while high order schemes are unstable and show oscillations in such areas.