Week 9 - FVM Literature Review

Finite Volume Method(FVM):-

• It is a method of representation for evaluating the PDEs in algebraic form.
• Here, Volume intergral is first converted into surface integral with the help of Divergence Theorem and hence solved further
• After getting into surface integral form this is solved calculating total flux entering and leaving the finite volume
• Here, a small volume termed as finite volume is used for all sort of calculations making a node at its centre
• Even if the structure is nonlinear we can mesh it easily using FVM

  Finite Difference Method	Finite Volume Method
  Written in Differential form of the PDE	Written in Integral form of PDE
  Each derivative is replaced by approximate difference formula	The PDE is rearranged in the form that it can be solved using finite volumes
  No requirements of structured grids to solve the problem	Requirement of structured grid to solve the problem
  The solution is obtained at each nodal point without flux conservation	The solution is obtained after flux conservations
  Can not compute discontinous solution	Can compute discontinous solution

Interpolation Schemes: Sometimes Volume integral requires values of the variables at the location other than the computational nodes of CV. To obtained values at these locations , Interpolation schemes are being used.

Example: 1D heat conduction Equation:

`del(alpha*(delT)/(delx))+S=0`

Here, the `alpha`and S both are dependent on temperature.

After solving it through FVM the final equation obtained is:

`alpha*A*[(T_e-T_p)/(Deltax)]-alpha*A*[(T_p-T_w)/(Deltax)]+Sourceterm`

LINEAR INTERPOLATION:-

We can approximate the value of the variable at the CV face centre by linear interpolation of the values at the two nearest nodes

At location 'e' the variable value is approximated as:-

`f_e=f_Elamda_e + f_p(1-lamda_e)`

where ,`lambda_e=(x_e-x_p)/(x_E-x_p)`

This can also be compared with central difference schemes in FDM , so it is also called as Central Difference Schemes(CDS)

Flux Limiter :

• It is used in high resolution schemes to avoid the spurous oscillation that would otherwise occur with high order spatial discretization schmes due to schocks, discontinuities or sharp change.
• The main idea behind this is to make the spatial derivative to realistic so that it can be made more physically realizable
• It operates only on sharp changing wavefronts, not the smooth one.

For the scheme given below:-

`(du_i)/dt+1/(Deltax_i)[F(u_(i+1/2))-F(u_(i-1/2))]=0`

`F(u_(i+1/2))` and `F(u_(i-1/2))` represents low and high resolution schemes respectively.

The flux limiter switching can be done as follows:

`F(u_(i+1/2))`= `f_(i+1/2)^(low)-phi(r_i)(f_(i+1/2)^(low)-f_(i+1/2)^(high))`

`F(u_(i-1/2))`=`f_(i-1/2)^(low)-phi(r_(i-1))(f_(i-1/2)^(low)-f_(i-1/2)^(high))`

f(low) is low resolution flux and f(high) is the high resolution flux

`phi(r)` is the flux limiter function

`phi(r)>=0`

So, when it is zero low resolution and when 1 it is high resolution