FVM literature review

The finite volume method (FVM) is a method for representing and evaluating partial
diffrential equation in the form of algebraic equations. This method is apply for over the
entire volume & properties are assumed to be concentrated at geometric centre of the
volume concerned. Actually finite volume method there is use intergral conservative
equation & its evaluated by using function value at compuational node.

Step to solve FVM problem
1. Find out integral form of equation
2. Apply gauss divergence theorem
3. Come up with suitable method to calculate flux term.
Type of scheme
1. Gradient scheme
2. Interpolation scheme

1. Gradient scheme
In gradient scheme for finding the temperature at east face e, then we take the
information neighboring point E & this is called gradient scheme.

2. Interpolation scheme

The approximation of surface & volume integral require values of the variable at location
other than the computational node of control volume in that case there is use of
interpolation scheme

Types

1. Upwind scheme - most commonly used for convection problems
2. Linear interpolation- Same as central difference scheme
3. Quadratic upwind interpolation
4. Hybrid scheme
5. Total variation Diminish (TVD)

Flux limiter need:

In order to capture shock & discontinuity that arises in hyperbolic equation, in physically
this equation model the convective fluid flow. So, it is has been observed that the loworder
scheme are usually stable but quite disspative in nature around the point of
discontinuity or shocks. On other side higher order numerical scheme are unstable in
nature & show oscillation in the vicinity of discontinuity, such behavior by analyzing the
modified equation of these scheme.
The main objective of flux limiter is to have highly accurate & stable oscillation free
scheme, this scheme called high resolution scheme.