Week 9 - FVM Literature Review

Objective:-

In your own words, describe the need for interpolation schemes and flux limiters in FVM. You can use the 1D linear heat conduction equation as an example.

Solution:-

What is Finite Volume Method:-

The finite volume method (FVM) is a method for representing and evaluating partial differential equation 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 computational fluid dynamic packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.

considering 1D linear heat equation:-

`frac{del}{delx}(alphafrac{delT}{delx})+s=0`

where,

s = source term

`alpha` = thermal diffusivity

consider a small volume 'dv' in the contol volume P, the integral form of the equation for small volume is given by:-

`int[del/(delx)(alpha*(delT)/(delx))+S]dV=0`

`intdel/(delx)(alpha*(delT)/(delx))dV+intSdV=0`

we know that, dV = A. dx

`int_w^edel/(delx)(alpha*(delT)/(delx))Adx+intSdV=0`

`(alphaAfrac{delT}{delx})^eunderset(w)( )V+S'dV=0`

where S' is the average value

`alpha_eA(delT)/(delx_e)-alpha_wA(delT)/(delx_w)+S'dv=0`

Assuming `alpha` as constant

alphaA[frac{Tunderset(e)( )-Tunderset(p)( )}{Deltax}]-alphaA[frac{Tunderset(p)( )-Tunderset(w)( )}{Deltax}]+S'dV=0

`|Heat Flux|underset(out)()-|Heat Flux|underset(text(in))()+Source=0`

Need for interpolation scheme:-

To calculate the `T_w` and `T_e`, temperature at W and e.

different types of interpolation scheme

Upwind differencing scheme

Central differencing scheme

Quadratic differencing scheme

Upwind differencing scheme

It is a first order accurate scheme and is numerically diffusive with a coefficent of numerical diffusion.

Approximation is done by the upper or next node of a control volume.

Central differencing scheme

It is a second order accurate scheme. approximation is done by the two nearest node of a control volume.

Quadratic differencing scheme

It is a third order accurate scheme. It is also represented as the QUICK interpolation.

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 partial differential equations (PDE's). They are used in high resolution schemes, such as the MUSCL 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).

How limiters work:-

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.

Need of Limiters:-

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 scheme. 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.