Week 9 - FVM Literature Review

 FINITE VOLUME METHOD

In this method we discretize a given equation into different volumes and integrate it for a volume dv. In other words it is a method for representing and evaluating partial differential equations in the form of algebraic equations. 

In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals using the divergence theorem.

For eg: `del/(delx)((alphadelT)/(delx))+S = 0`is the 1D linear heat conduction equation

The above equation is solved using the finite volume method by taking the integral of the equation for the whole volume i.e `intdel/(delx)((alphadelT)/(delx))dV+intSdV = 0`.

Here we calculate the integral of all the volumes present inside the control volume which results in the creation of the heat flux at the entry and exit i.e

`alphaA[(T_E-T_P)/(Deltax)]-alphaA[(T_P-T_W)/(Deltax)]+S_(avg)dV = 0`

In the above equation we have two heat fluxes and average of the source terms which helps in the heat generation.

DIFFERENCE BETWEEN FDM AND FVM

The main difference between finite difference method and finite volume method is that in FDM the partial differential equation is discretized into nodes or grid points whereas FVM the discretisation is done into different volumes and the average value inside the volume is calculated and the partial differential equation is solved.

In other words  the finite different method, which approximate derivatives using nodal values and  create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for the average value of the solution over some volume, and uses this data to construct approximations of the solution within cells

NEED FOR INTERPOLATION

Interpolation is the usage of known values to find unknown values.  Interpolation is achieved by using other established values that are located in sequence with the unknown value.

 Interpolation schemes are used to transform cell-centre quantities to face centres. The operation is used in many of the finite volume calculations, e.g. for the calculation of gradient, divergence, and Laplacian terms.

FLUX LIMITERS IN FVM

In FVM we have a scheme called gradient scheme which is a scheme that is used when we are trying to numerically represent the gradient of the temperature`(delT)/(delx)`,from which point we should use the information and how the scheme should be structured. This is called as gradient scheme.

While calcuating gradients we need to have the values bounded, thus we use Flux limiters in FVM. For eg if we have a shock flow problem there is a huge discontinuity and while calculating gradients the gradients will turn out to be huge values to limit these values from getting higher and to maintain a certain range of values of heat fluxes flux limiters are used .