Week 9 - FVM Literature Review

FDM is the oldest method for numerical solutions of PDEs. 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. The sentiment of the following comment is often seen in the literature. “This is a potential bottleneck of the method when handling complex geometries in multiple dimensions. 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. Extending to higher-order accuracy is very simple.

 

FEM has an important advantage in being able to deal with arbitrary geometries. The principal drawback, shared by any method that uses unstructured grids, is that the matrices of the linearized equations are not as well structured as those for regular grids making it more difficult to find efficient solution methods.

FEM is ideal for linear PDEs, expensive and complex for non-linear PDEs. Here higher order accuracy is achieved by using higher-order basis (i.e) shape functions. Extending to higher-order accuracy is relatively complex than FVM and FDM. Higher-order accurate calculations are expensive in computation and Mathematical formulation, especially for non-linear PDEs. Mostly suitable for Heat transfer, Structural mechanics, vibrational analysis, etc. In FEM, the differential equation some times conservative law may be violated

 

FVM can accommodate any type of grid, so it is suitable for complex geometries. 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. Since we have integrated the differential equation discretization is mathematically a valid one. It can be loosely viewed as FEM but the weight used in FVM is 1. In FVM, fluxes are integrated and the resultant is set to zero, so flux is conserved. FVM handles almost any PDEs and complex domain. Interpolation is done from face to center will reduce the accuracy of this process. FVM accuracy is based on the order of the polynomial used. FVM can also produce any order accurate numerical solution similar to FDM.

 

Finite Difference Method: It is difficult to satisfy conservation and to apply for irregular geometries

Finite Volume Method: It tends to be biased toward edges and one-dimensional physics.

Finite Element Method: It is difficult to solve hyperbolic equations using FEM.

 

A commercial package such as FLUENT uses Finite Volume Method.

 

Interpolation Schemes in Finite Volume Method (FVM):

The interpolation schemes used in FVM method are as follows

1. Upwind Interpolation Scheme. (UDS)
2. Central Differencing Approximation. (CDS)
3. Quadratic Upwind Differencing Scheme. (QUICK)
4. Hybrid Interpolation Scheme. (CDS & UDS)
5. Total variation Diminishing Scheme. (TVD)
   
   

1. Upwind Interpolation Scheme(UDS):

In this scheme we use the node in the upwind or upstream of the given node. It is generally used for convection dominated problems (i.e Pe>1). ,

Upwind schemes are the approximations that satisfy the boundedness unconditionally and is First order accurate in space.

 

2.Central Differencing Approximation (CDS):

Approximate values of the variable at the control volume (CV) face centre by the linear interpolation of the values at the two nearest computational nodes.

The linear interpolation is equivalent to that of using central difference formula of the first order derivative and hence this scheme is also termed as Central Differencing Scheme(CDS).

 

3. Interpolation Polynomial:

The approximation is second order precise. It is the simplest and most widely used interpolation technique for the evaluation of gradients required to measure diffusive fluxes.

FLUX LIMITERS:

Flux limiters are used in the numerical schemes to solve problems of fluid dynamics, described by highly coupled non-linear Partial Differential Equations. If we tend to use low order numerical schemes for the solving of the governing equations then we might get a highly oscillatory and unstable solution near the discontinuity if present. Whereas if we tend to use higher-order numerical schemes to capture the phenomenon more accurately we tend to exploit the computational power unnecessarily. The main idea behind the use of the flux limiters is to limit the spatial derivative terms near the discontinuities to physical real values so as to maintain physical consistency. There are several types of flux/slope limiter functions. They come in handy when sharp wave-fronts/peaks are present. For smoothly changing waves/functions, the flux limiters do not operate and the spatial derivatives can be represented normally.

It is imperative to note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities.