Interpolation and Gradient Schemes

The finite volume method (FVM) is a method for representing and evaluating PDE in the form of algebraic equations . In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many CFD packages. \"Finite volume\" refers to the small volume surrounding each node point on a mesh.

In finite volume method for evaluation of surface and volume integrals in the control volume it requires values of unknown variables at locations other than computational node points of cv.so for getting these values at these points we need to apply some sort of interpolations scheme which have been described below.

taking 1 D heat conduction equ for determining thermal conductivity which is a function of temperature for calculating integrals over a volume it requires values at face location ,which is possible only by interpolation scheme .since values are only available at centre of control volume not on the interface volumes

FIRST ORDER DIFFERENCING SCHEMES

The first order differencingschemes are the oldest differencing schemes specifically designed to fix the stability problems of the discrete advection equation. They are simple, stable, easily implemented, and provide a smooth solution. Unfortunately this is due to their being excessively diffusive for problems with moderate to high Reynolds or P´eclet numbers, resulting in solutions that are both quantitatively and qualitatively different from the correct solution.
Whilst these schemes are no longer commonly used they are included for historical interest. In addition they form the basis of some deferred correction forms of higher order methods.

The First Order Upwind Scheme (FOU)

The method was originally developed from a comparison with boundary layer flow, where the transport equations are parabolic, with scalars being transported downstream by the flow, and so the value of a scalar at a face can be approximated by it’s value at the first node upstream of the face. Thus for the east face of a cell

The Hybrid Scheme

In order to combine the accuracy of Central Differencing with the stability of the First Order Upwind scheme, a hybrid of the two schemes was suggested

The Exponential and Power Schemes

The scheme uses an exponential approximation to the scalar field and is called the exponential scheme, although it is sometimes referred to as the exact scheme

Second Order Differencing Schemes

The second order schemes are less commonly used than their first and third order counterparts. The simplest scheme is the previously discussed central difference method, which is not commonly used due to it’s problems with numerical stability. To overcome these limitations other second order schemes have been developed that use upwinding, with the value of a scalar at the cell face being extrapolated from two upstream points rather than using just one point as is the case with first order upwinding.

Central Difference Scheme (CDS)

Second Order Upwind Scheme (SOU)

Monotonic Second Order Upwind Differencing Scheme (MSOU)

Third Order Differencing Schemes

QUICK

As with the second order upwind schemes, third order upwinding increases the number of points in the computational molecule over that of the first order and central differencing schemes. In addition it can result in systems that are not diagonally dominant, and so it is normally recast into a deferred correction form.

B.FLUX LIMITERS

Flux limiters are used in high resolution schemes– numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by PDE\'s. They are used in high resolution schemes to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD).

and

represent edge fluxes for the ith cell. If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the gradients close to the particular cell, as follows,

,

where

low precision, high resolution flux,

high precision, low resolution flux,

flux limiter function,

represents the ratio of successive gradients on the solution mesh, i.e.,.