A REVIEW ON INTERPOLATION AND GRADIENT SCHEMES OF THE FVM METHOD

INTERPOLATION

1. In mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
2. In FVM, interpolation schemes are utilized to obtain the values of volume integrals required AT THE POINT OTHER THAN NODES.
3. Examples of interpolation schemes are UDS, CDS, QUICK etc.

UPWIND INTERPOLATION (UDS)

The upwind interpolation (UDS) for approximating the value of a variable at the east face of a control volume is given by

                              `f_e=f_P if (v.n)_e>0`

                              `f_e=f_E if (v.n)_e<0`

This interpolation scheme is equivalent to using a backward or forward finite difference approximation (depending on the flow direction). It is first order accurate, and is numerically diffusive with a coefficient of numerical diffusion `τ^(num) =rho*(∆x)/2`

LINEAR INTERPOLATION (CDS)

We can approximate the value of the variable at CV face centre by linear interpolation of the values at two nearest computational nodes. Thus, at location ‘e’ on a Cartesian grid, the variable value is approximated by

The linear interpolation is equivalent to the use of central difference formula of the first order derivative, and hence, this scheme is also termed as central difference scheme (CDS). This scheme is second order accurate, and may produce oscillatory solutions.

QUADRATIC UPWIND INTERPOLATION (QUICK)

A quadratic upwind interpolation (QUICK) scheme can be derived using polynomial fitting. The QUICK interpolation on a uniform Cartesian grid is given by

                   `∅_e=6/8 (∅_U )+3/8 (∅_D )-1/8(∅_(UU))`

where D, U and UU denote the downstream, the first upstream and the second upstream node respectively.

QUICK scheme delivers solutions with third order accuracy but is prone to oscillations.

FLUX LIMITERS

1. As the name suggests, they LIMIT the spatial derrivatives to REALISTIC VALUES that are physically reliable. 
2. Usually used in higher resolution schemes for solving problems as described by the partial differential equations.
3. Flux Limiters are mostly utilized in case a sharp wavefront is present.
4. In its essence, a flux limiter would be helpful in avoiding wiggles (or spurious oscillations) that occur while the usage of higher order spacial discretized schemes due to shocks, sudden or abrupt changes or any sort of discontinuity in the pool of the solution values.
5. The following INTERNET PICTURE depicts a few examples of limiters used in CFD.

CONCLUSION

The FVM interpolation methods and limiters have been studied in detail.