FVM Literature Review

Finite Volume Method (FVM)

• is based on the integral form of conservation equations.
• The problem domain is divided into a set of non-overlapping control volumes.
• The conservation eqns are applied to each finite volumes.
• The integrals occuring in the conservative equations are evaluated using functions values at computational nodes (Usually taken as centroids of CVs)
• This process involves use of approximate integral formulae and interpolation methods to obtain the values of variables at surfaces of the CVs
• FVM can accommodate any type of grid and hence it popularaties in CFD commercial pakages.

Interpolation Methods.

• The approximations of surface and volume integral requires values of the variable at locations other than the computational nodes of the CV.
• Values at these locations are obtained using interpolation formulae.
• Various possibilities for interpolation, e.g.,
  • Upwind interpolation
  • Linear interpolation
  • Quadratic upwind interpolation
  • Hybrid, TVD and ENO interpolation schemes ...

Flux Limiter

Flux limiters schemes are used to limit the spatial derivatives to realistic values for scientific and engineering problems this usually means physical realisable and meaningful values and it comes into picture when sharp wave fronts are present. Consider a 1D semi-discrete scheme below

`(du_i)/dt + 1/(Deltax_i)[F(u_(i+1/2)) - F(u_(i-1/2))] = 0`

where `F(u_(i+1/2))` and `F(u_(i-1/2))` 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 cells as follows,

`F(u_(i+1/2) )= f_(i+1/2)^(low) - phi(r_i) (f_(i+1/2)^(low)-f_(i+1/2)^(high))`

`F(u_(i-1/2) )= f_(i-1/2)^(low) - phi(r_(i-1)) (f_(i-1/2)^(low)-f_(i-1/2)^(high))`

Where,

`f^(low) =` low precision, high resolution flux,

`f^(high) =` high precision, low resolution flux,

`phi(r) =` flux limiter function,

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

`r_i = (u_i - u_(i-1))/(u_(i+1)-u_i)`

The limiter function is constrained to be greater than or equal to zero. 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.

Some of the Limiter functions are CHARM, HCUS, HQICK, Koren, minimod, monotonized central (MC), Osher, ospre and list goes on.