Interpolation Schemes and Flux Limiters

In Finite volume method the computational domain is subdivided into finite number of control volumes where the boundaries define the Control Volume and the centres of these are the computational node.

Considering !D conduction:

Using the Integral form:

`intd/dx(gamma(dphi)/(dx))dV+intSdV=0`

`(gammaA(dphi)/(dx))_e-(gammaA(dphi)/(dx))_w+SDeltaV=0`

`gamma_eA_e((phi_E-phi_P)/(Deltax_(PE)))-gamma_wA_w((phi_W-phi_P)/(Deltax_(PW)))+S_c+S_Pphi_P=0`

It can be seen that the above equation consists of terms containg values at e and w wich are the points on the walls. But in FVM all computational nodes are the cell centres. Hence it is required to interpolate these values to obtain the values of the parameters at the walls of the control volumes. Interpolation Schemes are basically the equations where the cell face data is written as a function of the cell centre data(i.e computational node).The source term is linearised.

Consider the one dimension convection diffusion equation:

`d/dx(rhouphi)=d/dx(gamma(dphi)/(dx))`

the continuity equation is also to be solved along with this

`d/dx(rhou)=0`

`intd/dx(rhouphi)=intd/dx(gamma(dphi)/(dx))`

`(rhouphi)_e-(rhouphi)_w=[gamma(dphi)/(dx)]_e-[gamma(dphi)/(dx)]_w`

`(rhouphi)_e-(rhouphi)_w=gamma_eA_e((phi_E-phi_P)/(Deltax_(PE)))-gamma_wA_w((phi_W-phi_P)/(Deltax_(PW)))`

`(rhou)_ephi_e-(rhou)_wphi_w=gamma_eA_e((phi_E-phi_P)/(Deltax_(PE)))-gamma_wA_w((phi_W-phi_P)/(Deltax_(PW)))`

The LHS in the above equation requires data from the cell faces and hencethis is obtained by interpolation schemes.

Upwind Scheme:In this scheme the direction of velocity of taken into consideration and the face value is chosen. If the velocity goes from west to east then:

`phi_w=phi_W and phi_e=phi_P`

If velocity direction is from east to west:

`phi_e=phi_E and phi_w=phi_P`

`(rhou)_ephi_E-(rhou)_wphi_P=gamma_eA_e((phi_E-phi_P)/(Deltax_(PE)))-gamma_wA_w((phi_W-phi_P)/(Deltax_(PW)))`

Comparing with the taylor series expansion it can be noted that only the constant term is considered here and the first order derivative onwards in neglected and hence this scheme has first order accuracy.

Central Differencing Scheme:

`phi_e=(phi_P+phi_E)/2 and phi_w=(phi_W+phi_P)/2`

`(rhou)_e[(phi_W+phi_P)/2]-(rhou)_w[(phi_W+phi_P)/2]=gamma_eA_e((phi_E-phi_P)/(Deltax_(PE)))-gamma_wA_w((phi_W-phi_P)/(Deltax_(PW)))`

Comparing with the taylors series, the second order derivative is neglected. Hence the CDS scheme has second order accuracy.

Power law:

`phi_e=phi_P-(1-0.1Pe)^5/(Pe)(phi_E-phi_P)`

Generalising

`phi_(face)=phi_(i-1)-(1-0.1Pe)^5/(Pe)(phi_i-phi_(i-1))`

Here Pe is the peclet number `Pe=(rhou)/(gamma/Deltax)=( CONVECTION)/(DIFFUSION)`

When Peclet number is equal to 10 it reduces to Upwind scheme.

Second Order UDS:

Here the value of fluxes are determined from the cell values in the two cells upstream of the face. Even though this scheme is more accurate than the first order upwind method, in regions where the gradients are strong it can results in unrealistic values which are not in range of the computational domain. Hence Flus limiters may be used in these cases.

QUICK Scheme:

QUICK stands for Quadratic Upwind Interpolation for Convective Kinetics.

This is similar to the Second order Upwind Method.However, the data is taken from two upstream nodes and one downstream node.A quadratic curve is fitted through these nodes. Even here the sceme is accurate but in regions of strong gradients, the values can overshoot or undershoot and lead to instabilities.

`phi_e=6/8phi_P+3/8phi_E-1/8phi_W`

generalising

`phi_(face)=6/8phi_(i-1)+3/8phi_i-1/8phi_(i-2)`

Flux Limiters:

Flux limiters are used in high resolution schemes. They limit the solution gradient near shocks and discontinuities. They basically act like switches and ensure that the reconstruction of slopes do not exceed above or dip below the value on the adjacent cells.

The reconstruction equation is as follows:`u_i+(Deltax)/2(Slope)`

`u_i+(Deltax)/2(u_i-u_(i-1))/(Deltax)phi(r)`

where `phi(r)` is the flux limiter

`u_i+(Deltax)/2(u_i-u_(i-1))/(Deltax)phi(r) <= u_(i+1)`

`u_i+(Deltax)/2(u_i-u_(i-1))/(Deltax)phi(r) >= u_(i-1)`

`phi(r)`helps satisfy these conditions

`r=(u_(i+1)-u_i)/(u_(i)-u_(i-1))`

When

`(u_(i+1)-u_i)=(u_(i)-u_(i-1))` then `phi(r)=1`

i.e. r=1 `phi(r)=1`

When

`(u_(i+1)-u_i) and(u_(i)-u_(i-1))` are of opposite signs then `phi(r)=0`

i.e r<0 `phi(r)=0`

The Green region shows the viable region of the flux limiters

Different Flux Limiters are used along with different higher order schemes and the range of each of Flux limiter changes accordingly.

Some examples for flus limiters are Minmod limiter, Vanleer, etc.