Week 9 - FVM Literature Review

Finite Volume Method

The Finite Volume Method (FVM) is a discretization method for the approximation of a single or a system of partial differential equation expressing the conservation, or balance, of one or more quantities. It is based on approximate solution of integral form of conservation equations. The problam domine is divided into a set of non-overlapping control volumes (called finite volumes). The conservation equations are applied to each finite volume. Integrals occurring in the conservation equations are evaluated using function values at computaional nodes.

Differences between FDM & FVM

FDM is very defficult to use when the grid are not allingned in x,y & z directions. And it is tough to use with boundary-condition problem on curved boundaries. FDM is more suitable for rectanguler or quadrilateral shapes. But in FVM instead of writing the equation in points, we write it for a volume. Then we apply few mathamatical tecniques to help us solve the equations for any type of grid. FVM preserves the consept of conservations.

Interpolation schemes

The use of interpolation schemes to approximate the property values located at the boundaries of the control volumes are necessary when FVM is applied, using as interpolation nodes the correspondent properties located at the center of the adjacent control volumes. Interpolation schemes demand the use of a high degree of refinement of the meshes due their lack of accuracy.

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 Monotonic Upstream-centered Scheme for Conservation Laws (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). It has the effect of limiting the solution gradient near shocks or discontinuities.

1D linear heat conduction equation

P = central point of control volume

W = west point

E = east point

w = west connecting face

e = east connecting face

S = source term (responsible for heat generating)

A = Face area

`del/(delx) * (alpha (delT)/(delx)) + S = 0` -> conservation equation

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

`int_w^e(del/(delx) * (alpha (delT)/(delx))dv) + intS dv = 0`

`int(del/(delx) * (alpha (delT)/(delx)) A dx + intS dv = 0`

`int del/(delx) (f) dx = f`

`[alpha A (delT)/(delx)]_w^e + barS dv = 0`

`underset(e)(alpha) A * (delT)/(delunderset(e)(x)) - underset(w)(alpha) A * (delT)/(delunderset(w)(x)) + bar(S)dv = 0`

`[ alpha A [ (underset(E)(T) - underset(P)(T))/(Deltax)]] - alpha A [ (underset(P)(T) - underset(W)(T))/(Deltax)]] + barS dv = 0`