FVM

Finite Volume Method (FVM)

Finite volume method provides a robust way of discretization the governing equations to solve for the certain class of heat transfer and fluid flow problems.Similar to the FDM technique wherein derivatives are replaced/dicretized in form of differences that is been applied on the points/nodes , FVM applies the same concept but the derivatives which are discretized are applied to the volumes(with centres).In short , FVM computes solution at each finite volumes where FDM computes solution at each points .Also FVM seems to conserve the properties in a better way than the FDM because of the fact that the governing equations are solved in an intergal form .On the other hand the Finite Element Method technique is also a popular technique that is been used in many structural simulation commerical codes , Its usgae in CFD is limited because it could not able to model the turbulence

Interpolation schemes.

The main purpose of these schems serves in finding the intial value for the boundary that actually anchors the solution .In the heat conduction equations you could find the thermal diffusivity terms in each of the heat flux constituents.To evaluate this property we need to evaluate temperature at that location first and that requires some interpolation schemes to obtain the appropriate value of the quantity at the particular location/boundary/face of the volume respectiviely .

Let us look at various interpolation schemes that a FVM uses .

UPWIND INTERPOLATION SCHEME

This technique is similiar to the Forward or backward difference schemes .It is first order accurate and is numerically diffusive

Quadratic upwind Interpolation scheme

This method can be derived using polynomial fitting .This is a third order accurate but it is prone to fluctuations that results in solution instability.

Linear Interpolation scheme

We can approximate the value of the quanity at the control volume centre by linearly interpolating the values at nearest computation grid.This scheme is something similiar to a central difference scheme .This is a second order accurate but it also prones to solution instability .

Flux limiters :

Whenever there is a numerical representation with which gradient of a particular quantity is to be determined ,we ensure that the gradient should be bounded (meaning it should jump) .This happens whenever there are discontinuities in flow (as in shock flow problems) that leads to a gradient value that is higher than it shold be .To prevent this , flux limiters are being used both for the Lower and higher order approximations to bound the values of gradient such that it maintains in a certain limit .

The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values – for scientific and engineering problems this usually means physically realisable and meaningful values. They are used in high resolution schemes for solving problems described by PDEs and only come into operation when sharp wave fronts are present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations.