Week 9 - FVM Literature Review

Finite Volume Method(FVM):

• The Finite volume method(FVM) is a method for representing and evaluating partial differential equations in form of algebraic equations.
• In this method, volume integrates into a partial differential equation that contains a divergence term and is converted into 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.
• 'Finite volume' refers to the small volume surrounding each node point on a mesh.
• The advantage of the finite volume method is that it is easily formulated to allow unstructured meshes and this method is used in many computational fluid dynamic packages.

Differences between FDM & FVM:

• A finite difference method(FDM) discretization is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula that can generally be derived from a Taylor series expansion. The computational domain is usually divided into hexahedral cells(grid) and the solution will be obtained at each nodal point.
• A finite volume method(FVM) discretization is based upon an integral form of PDE to be solved(eg. conservation of mass, momentum, or energy). The PDE is written in a form that can be solved for a given finite volume for cells. The computational domain is discretized into finite volumes and then for every volume, the governing equations are solved. The resulting system of the equation usually involves fluxes of the conserved variables, and thus the calculation of fluxes is very important in FVM
• The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the given mesh into a structured numerical grid internally is completely avoided

INTERPOLATION SCHEMES:
The approximation of surface and volume integrals may require values of the variable at locations other than the computational nodes of the CV. Values at these locations are obtained using interpolation formulae.

UPWIND INTERPOLATION:

The upwind interpolation (UDS) for approximating the value of a variable at the east face of
a control volume is given by;

fe=fpif(v.n)e>0;feif(v.n)e<0fe=fpif(v.n)e>0;feif(v.n)e<0

This interpolation scheme is equivalent to using a backward or forward finite difference approximation (depending on the flow direction).

LINEAR INTERPOLATION (CDS):
We can approximate the value of the variable at the CV face center by linear interpolation of the values at the two nearest computational nodes. Thus, at location 'e' on a Cartesian grid, the variable value is approximated by;

fe=fEλe+fp(1−λe),fe=fEλe+fp(1-λe),

λe=xe−xpxE−xpλe=xe-xpxE-xp

The linear interpolation is equivalent to the use of the 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.

QUADRATIC UPWIND INTERPOLATION (QUICK):
A quadratic upwind interpolation (QUICK) scheme can be derived using polynomial fitting. The QUICK interpolation on a uniform Cartesian grid is given by;

ϕe=68ϕU+38ϕD−18ϕUUϕe=68ϕU+38ϕD-18ϕUU

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). The QUICK scheme is third-order accurate, but prone to oscillations.

Flux limiters:

Flux limiters are used in the high-resolution scheme used to solve the problem in science and engineering, particularly fluid dynamics described by PDE's. They are used in a high-resolution scheme such as the MUSCL scheme to avoid the spurious oscillation that would otherwise occur with high order spatial discretization scheme due to shocks, discontinuities, or sharp changes in the solution domain. The use of flux limiters together with an appropriate high-resolution scheme makes the solution's total variation diminishing (TVD).

The main idea behind the construction of flux limiters schemes is to limit the spatial derivatives to realistic values for scientific and engineering problems this usually means physically realizable and meaningful value. They are used in the high-resolution scheme for solving problems described by PDE's and only come into operation when sharp wavefronts are present. For smoothly changing waves, the flux limiters do not operate, and the spatial derivatives can be represented by high-order approximation without introducing spurious oscillations.

Consider the 1D semi-discrete scheme below;

where;

F(ui+12)andF(ui−12)F(ui+12)andF(ui-12) represents low and high-resolution schemes, then the flux limiters can be switch between these schemes depending upon the gradient close to a particular cell as follows;

where,

flowflow= low-resolution flux

FhighFhigh= high-resolution flux

ϕ(r)ϕ(r)= flux limiter function and r represent the ratio of successive gradients on solution mesh.

ri=ui−ui−1ui+1−uiri=ui-ui-1ui+1-ui

The limiter function is constrained to be greater than or equal to zero. i.e ϕ(r)≥0ϕ(r)≥0. Therefore, when the limiter is equal to zero the flux is represented by a low-resolution scheme. Similarly, when the limiter is equal to 1 it is represented by a high-resolution scheme. The various limiters have differing switching characteristics and selected according to the particular problem and solution scheme.

REFERENCES:

1. Versteeg, H. K. and Malalasekera "Introduction to Computational Fluid Dynamics: The Finite Volume Method"
2. https://www.youtube.com/watch?v=WL75T7iCSaA&feature=emb_logo