Week 9 - FVM Literature Review

Finite Volume Method:-

It is one of the method for evaluation and representation of differential equations in form of algebraic equations. This method can be used on all differential equations which can be written in the divergence form. The integration is done over the volume. Applying gauss theorem, the volume integral over the divergence can be converted into surface integral across the boundaries, which fascilitates surface integrals of the fluxes of the dependent variable across the boundary of the cells.
As, FVM looks upin fluxes of the dependent variable across the boundary of the cells, a conservation approach of the dependent variable is achieved.

Differences between Finite Difference and Finite Volume Method:-

The main difference between these two methods are, how they are applied. Finite difference cant be used, when the grid points are not alligned along x,y or z direction. Finite volume method solves this issue as instead of writing the equation for a node point, it is written for a volume.

In Finite volume method, the mathematical technique are applied to solve the equation for any type of grid. Also, the finite volume method preserves the concept of conservation.

Need for Interpolation:-

1-D heat condution:-

Let the domain be divided as displayed above. Each subdomain is represented by a green grid point at the center. Let the middle cell be be P, left to it is W and right ot it is E. The center of the cell represent the average of properties of the flow. Adjacent face between cell W and P is w. The adjacent face between P and E is e.

The approximations of surface and volume integral when required at places other than node points, then the interpolation method is used.

The governing equation of 1-D heat equation with source term :-

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

let `barS` be the average source term. With volume integral,

`int{del}/{delx}(alpha{delT}/{delx})dV + intbarSdV = 0`

Since `dV = Adx`

`int{del}/{delx}(alpha{delT}/{delx})Adx + intbarSdV = 0`

Applying upper and lower limit

`[alphaA{delT}/{delx}]_w^e + barSdV = 0`

`alpha_eA{delT}/{delX_e} - alpha_wA{delT}/{delx_w} +barSdV = 0`

Taking `alpha` constant, using gradient scheme for face temperature deravitives,

`alphaA[{T_e-T_p}/{Deltax}] - alphaA[{T_p-T_w}/{Deltax}] + barSdV = 0`

Here,

`alphaA[{T_e-T_p}/{Deltax}]` and `alphaA[{T_p-T_w}/{Deltax}]` are the heat flux.

In order to approximate the dependent variable (T in this case) at w, and e face, we need to use the interpolation scheme.

Some of the types of interpolation schemes are:-

1. Upwind Interpolation

2. linear Interpolation

3. Quadratic upwind Interpolation

Flux limiters:-

Selection of appropriate gradient scheme becomes necessary, as they should be bounded for the solution to be stable. Flux limiters are used to avoid the oscillations that occurs with high order spacial discretization schemes due to shocks.