FVM Interpolation and Gradient Schemes literature review

FVM-Flux limiters and Interpolation Schemes

What is FVM?

FVM is a Finite Volume method for representing and evaluating PDE\'s in the form of algebraic equations, which are similar to FDM or FEM.values are calculated at discrete places on a meshed geometry. Finite Volume refers to the small volume surrounding each node point on a mesh. In this 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.

This method is helpful if the mesh is not even or unstructured meshes , like a mesh we give in OpenFoam using simpleGrading which is not possible in Matlab..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 .

So while writing the Governing equation in intergral form we come up with suitable methids to calculate Flux terms.In this process we use interpolation and gradient schemes. We use Flux limiters to bound the values.

1.Interpolation Schemes:

The approximation of surface and volume integrals may require values of the variable at locations other than the computational nodes of the CV. Values at these locations are obtained using interpolation formulae.

For Example in 1D Linear Heat conduction equation in steady state , the source and alpha values are function of tempertaure , we need to calculate them seperately of temperature to that particular face. To calculate the temperature at that face, we use interpolation scheme which takes the temperatures from adjoining cells.

We mention few of the possibilities,

a.Upwind Interpolation:

It is used for calculating approximate value of wariable at the east face of control volume, it is first order accurate and is numerically diffusive with coefficient of numerical diffusion.

It is equivalent to using a backward or forward finite difference approximation. The upwind relation is given below. e stands for east side.

b. Linear Interpolation:

To approximate the value of variable at control volume face center by two nearest computational nodes location e.

It is equivalent to central Difference schemes of the first order derivative. It is second order accurate and produce oscillatory solutions. the relation for linear interpolation is given below.

c. Quadratic Upwind Interpolation:

It is third order accurate but prone to oscillations it is derived from polynomial fittings. The QUICK interpolation on a uniform cartesian grid is given by

where D,U and UU denote the downstream , the first upstream and second upstream node respectively(E,P and W or P,E and EE depending on the flow direction).

A quadratic curve is fitted with two upstream nodes and one downstream node. But in regions with strong gradients , overshoots and undershoots can occur.This can lead to stability problems in the calculation.

The other interpolation schemes are:

d.Hybrid Interpolation Schemes: (CDS and UDS)

e. Total Variation Diminishing (TVD) Scheme

2.Flow 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, such as the MUSCL scheme, 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).

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. Consider the 1D semi-discrete scheme below,

where F(u i+1/2) and F(u i-1/2) represents 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,

The limiter function is constrained to be greater than or equal to zero, i.e.. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradient), the flux is represented by a low resolution scheme. Similarly, when the limiter is equal to 1 (smooth solution), it is represented by a high resolution schemes. The various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, and a particular choice is usually made on a trial and error basis.