Week 9 - FVM Literature Review

Finite Volume Method:

The Finite Volume Method (FVM) is a numerical technique that transforms the partial differential equations representing conservation laws over differential volumes into discrete algebraic equations over finite volumes (or elements or cells).

Major differences between FDM & FVM:

• On structured grids, the FDM is very simple and effective. It is especially easy to obtain higher-order schemes on regular grids. The disadvantage of the FDM is that certain conservation laws are not enforced unless special care is taken. Furthermore, the restriction to simple geometries is a significant disadvantage if complex
  flows are specified.
• Furthermore, the FDM uses a topologically square network of lines to construct the discretization of the PDE.
• FDM is an easy method but not reliable for conservative differential equations and solutions having shocks. Tough to implement in complex geometry where it needs complex mapping and mapping makes governing equations even tougher.
• FVM can accommodate any type of grid, so it is suitable for complex geometries. FVM handles almost any PDEs and complex domain.
• The disadvantage of FVMs compared with FD schemes is that methods of order higher than second are more difficult to develop in 3D.
• FVM is similar to FDM but the method (FVM) integrates the differential equation over a control volume and discretizing the domain.
• FVM accuracy is based on the order of the polynomial used. Interpolation is done from face to center will reduce the accuracy of this process.
• So one can infere that, in FDM, it is difficult to satisfy conservation and to apply for irregular geometries and, in FVM, it tends to be biased toward edges and one-dimensional physics.

If we consider 1D linear heat conduction equation i.e. `del/(delx)(alpha(delT)/(delx))+S=0`

In FVM, first step is the discretization of the geometric domain into non-overlapping elements or finite volumes.

As shown in above figure, geometric domain has been discretised into P, W and E elements. If we consider 'dv' volume from element P.

Second step - The partial differential equations are then discretized/transformed into algebraic equations by integrating them over each discrete element.

`int(del/(delx)(alpha(delT)/(delx))+S)dv=0`

`intdel/(delx)(alpha(delT)/(delx))dv+intS.dv=0`

If we consider `dv=A.dx`, no thickness along z-axis.

`intdel/(delx)(alpha(delT)/(delx))A.dx+intS.dv=0`

For the element P, it's boundaries are 'w' and 'e'. So, integration limits change as per this. Also, source term 'S' can be integrated by considering an average 'S' i.e. `barS`.

`[alphaA(delT)/(delx)]_w^e+barSdv=0`

`alpha_eA((delT)/(delx))_e-alpha_wA((delT)/(delx))_w+barSdv=0`

By using forward differencing for first order partial derivative, such as, `(delT)/(delx)=(T_u-T_l)/(Deltax)`

`alphaA[(T_E-T_P)/(Deltax)]-alphaA[(T_P-T_W)/(Deltax)]+barSdv=0`

Third step - The system of algebraic equations is then solved to compute the values of the dependent variable for each of the elements.

For element P, it's temperature at centroid is `T_P`. So, to calulate temperature at interface such as 'e', we need to use interpolation scheme. The interpolation schemes contains terms that are interpolations of values typically from cell centres to face centres. The specification of these schemes requires the name of the flux field on which the interpolation is based.

Flux limiters are used in high resolution schemes used to solve PDEs. 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).