FVM Interpolation Scheme and flux limiter

Interpolation Scheme:-

In Finite Volume Method,for evaluation of surface and volume integrals in a control volume,it require values of unknown variables at locations other than computational node points of CV.So for getting values at these unknown location we need to apply some sort of inerpolation scheme available like upwind,central difference Scheme etc.

Taking 1D Heat conduction equation,for determining Thermal Conductivity(alpha),which is a function of temperature,for calculating integrals over a control volume ,it requires values at face location,which is possible only by Interpolation scheme.Since values are only availale at centre of control volume not on interface of volumes.

Value of f is not available at cell face,we need to interpolate f,the only unknown in f is Φ

Different methods to approximate Φ are;

1.Upwind Differencing Scheme

2.Central Differencing Scheme

3.Quadratic Upwind Interpolation(QUICK)

1.Upwind Differencing Scheme:

2.Central Differencing Scheme:

3.Quadratic Upwind Interpolation(QUICK):

Flux Limiters:-

Flux limiters are used in numerical schemes to solve problems in fluid mechanics that are described by partial differential equation.

It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity.

Flux limiters function to construct high resolution schemes. The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of socks/discontinuities.Example-Total Variation Diminishing Scheme.

Limiter functions

The following are common forms of flux/slope limiter function;

1.Superbee

2.Sweby

3.HQUICK