Need for interpolation schemes and flux limiters in FVM approach.

In the FVM, interpolation schemes are used to determine the values of properties at cell faces after discretization.

Flux limiter functions are used with a class of discretization schemes called as Total Variation Diminishing schemes to achieve oscillation free solutions in CFD. These schemes were developed to obtain higher order wiggle free results from a CFD solution.

Both can be explained by considering the following 1D steady heat conduction example -

1. Governing equation -

`nabla.(Gamma (nablaphi)) +S_phi = 0`

where `Gamma`is the diffusion coefficient

`S_phi`is the source term (assumed to be depending on `phi`).

2. Control volume integration of the governing equation

This is the fundamental difference between FVM method and all the other methods.

`int_(CV)[nabla.(Gamma (nablaphi))] +int_(CV)[S_phi] = 0`

Using Gauss divergence theorem

`int_(A)dot n.[(Gamma (nablaphi))]dA +int_(CV)[S_phi] = 0`

3. Numerical Stencil -

The numerical stencil is

Here, A and B are the domain boudaries.

P is the current nodal point(cell center). E and W are its east and west neighbours respectively.

The face of the control volume are denoted by e and w for the east and west side respectively.

It is assumed that the boundaries of each CV is midway between adjacent nodes.

The zoomed in view of the highlighted CV is -

`deltax_(WP) and deltax_(PE)` are the distances between the nodal points while

`deltax_(wP) and deltax_(Pe)` are the distances between face w and node P and face e and node P respectively.

Thus the control volume width is

`Deltax = deltax_(we)`

4. Discretization of the Governing Equation

The 1D form of the equation becomes

`int_(CV)[d/dx(Gamma (dphi)/dx)] +int_(CV)[S_phi] = 0`

After discretation in the CV

`[(Gamma A(dphi)/dx)]_e - [(GammaA (dphi)/dx)]_w + bar (S_phi) DeltaV = 0`

A is the cross sectional area of the control volume

`bar(S_phi)` is the average value of the source term in the control volume.

The first and second terms in the above equation are called flux terms and must be evaluated at the control volume boundaries.

Different interpolation techniques have thus been developed to evaluate these values at the control volume faces by utilizing the values of property`phi` and `Gamma` at the neighbouring nodes E and W respectively.

For uniformly distributed points, the most common and simplest technique used is the central differencing scheme which gives

`Gamma_e = (Gamma_P + Gamma_E)/2`

`Gamma_w = (Gamma_W + Gamma_P)/2`

and the flux terms thus become

`[(Gamma A(dphi)/dx)]_e = Gamma_eA_e*(phi_E-phi_P)/2`

`[(Gamma A(dphi)/dx)]_w = Gamma_wA_w*(phi_P-phi_W)/2`

The source term is also linearized as

`bar(S_phi) = S_u + S_Pphi_P`

These values are then substituted in the discretized equation to obtain the final form of the govering equation as

`a_pphi_p = a_Wphi_W + a_Ephi_E + S_u`

where

`a_W`	`a_E`	`a_P`
`Gamma_w/(deltax_(WP))A_w`	`Gamma_e/(deltax_(PE))A_e`	`a_W +a_E - S_P`

There are many other interpolating schemes which are more accurate than the central differencing schem discussed above. Some of them are -

1. Upwind scheme (most stable i.e. wiggle free but only first order accuracte)
2. Hybrid scheme
3. Power law scheme
4. QUICK
5. TVD schemes

Each of the above as its own benefits and shortcomings. However, all the schemes must have the following properties -

1. Conservativeness
2. Boundedness
3. Transportiveness

To achieve the goal of higher order accuracy and stability, TVD (Total variation diminishing schemes) are used.

The higher order schemes can be generalized as

`phi_e = phi_P + 1/2psi(phi_E-phi_P)`

Here `psi` is an appropriate function

It can be derived for all the above interpolation schemes that

`psi = psi(r)`

with

`r = (phi_P-phi_W)/(phi_E-phi_P)`

r is the ratio of the upwind side gradient to downwind side gradient.

Except for upwind differnecing scheme, all the schemes are conditionally TVD.

This funciton `psi(r)`is thus called flux limiter and is very important for stable solutions.

Note - The book Introduction to CFD by H K Versteeg and W Malalasakera has been referred.