Week 9 - FVM Literature Review

Need of Interpolation schemes in Finite Control Volume Method

The approximation of surface and volume integral may require values of the variable at locations other than the computational nodes of C.V. So, values at these locations are obtained using Interpolation.

There are different Interpolation schemes such as :

• Linear Interpolation Scheme
• Upwind Interpolation Scheme
• Quadratic Upwind Interpolation Scheme
• Hybrid Interpolation Scheme

In Linear Interpolation Scheme (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.

The central-difference schemes are simpler to program and require less computer time per time step.

This scheme is more accurate than the first order upwind scheme if Peclet number is less than 2.

The Upwind Differencing Scheme is a method used in numerical methods in CFD for convection-diffusion problems. This scheme is specific for Peclet number greater than 2 or less than −2

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

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.

By taking into account the direction of the flow, the upwind differencing scheme overcomes that inability of the central difference scheme. This scheme is developed for strong convective flows with suppressed diffusion effects.

A Quadratic Upwind Interpolation (QUICK) Scheme can be derived using polynomial fitting. The QUICK interpolation on a uniform Cartesian grid is given by

Where D, U and UU denote the downstream, the first upstream and the second upstream node respectively (E, P, and W or P, E and EE depending on the flow direction). QUICK scheme is third order accurate, but prone to oscillations.

A Hybrid Difference Scheme is a method used in the numerical solution for convection-diffusion problems.It is a combination of Central Difference Scheme(CDS) and Upwind Difference Scheme(UDS) as it exploits the favorable properties of both of these schemes.

It makes use of the central difference scheme, which is second order accurate, for small Peclet numbers (|Pe| < 2). For large Peclet numbers (|Pe| > 2) it uses the Upwind difference scheme, which first order accurate but takes into account the convection of the fluid.

It produces physically realistic solution and has proved to be helpful in the prediction of practical flows. The only disadvantage associated with hybrid difference scheme is that the accuracy in terms of Taylor series truncation error is only first order.

Flux Limiters in FVM

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).

Generally used flux limiters :

CHARM

minmod

HCUS

HQUICK

superbee

Osher

van Leer

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 scheme 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.