Describing the need of Interpolation Schemes and Flux Limiter in Finite Volume Method

OBJECTIVE:

To describe the need for interpolation schemes and flux limiter in Finite Volume Method.

SOLUTION:

Interpolation scheme:

The approximation of volume and surface integrals may require values of the variable at locations other than the computational nodes at control volume.

There are different standard method for interpolation such as linear interpolation,quadratic interpolation,spline interpolation and so on. Depend upon the physical description of our problem suitable interpolation scheme will be used.

Here we take the example of 1D steady state heat conduction problem,

`(dT)/(dx)(k (dT)/(dx))+S=0`

STEP 1:

Divide the domain into a number of finite sized sub-domain (CV) each sub-domain is represent by finite number of grid point. we use only one grid point at centre which represent the averege properties of the flow.

STEP 2:

Integrate the governing differential equation over each sub domain

`\\int_{w}^e d/dx(k(dT)/dx)dx+\\int_{w}^e(a+bT)dx=0`

STEP3:

Consider a profile assumption for the dependent variable for evaluating the above integrals to express the results in terms of algebraic quantity at the grid points.

Now ,this is where interpolation schemes are used in FVM

here we choose the piecewise linear interpolation scheme is used to determine the temperature distribution along the faces of grid points(w and e).

`kundersete()(TundersetE()-TundersetP())/(deltaxundersete())-kundersetw()(TundersetP()-TundersetW())/(deltaxundersetw())+(a+bTundersetp())Deltax=0`

`((kundersete())/(deltaxundersete())+(kundersetw())/(deltaxundersetw())-bDeltax)Tundersetp()=(kundersete())/(deltaxundersete())TundersetE()+(kundersetw())/(deltaxundersetw())TundersetW()+aDeltax`

we can also write the above equation,

`aundersetP()xxTundersetP()=aundersetE()TundersetE()+aundersetW()TundersetW()+b`

we can write in the form of indices,

`aunderseti()Tunderseti()=aunderset(i+1)()Tunderset(i+1)()+aunderset(i-1)()Tunderset(i-1)()+b`

finally, we see that finding the variable of one particular grid point expressed by variable function of the neighboring grid points.

Flux Limiters:

For example, In a upwind scheme produces too much numerical viscosity,which leads to damage the shapes near the discontinuities and the second order scheme produces oscillations.Therefore we can say that the second order schemes has over corrected the first order schemes near the discontinuities,whereas in a smooth flow region the second order scheme gives better than do the first order schemes.The Flux Limiter method tries to limit the contribution from the second order correction near the discontinuity.