Week 9 - FVM Literature Review
FDM
Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that we do not want included in our simulation. With the finite-difference method, we may easily run into problems handling boundaries for the purpose of defining the boundary conditions.
The finite-difference method is the most direct approach to discretizing based upon the differential form of the PDE to be solved. We consider a point in space where we take the continuum representation of the equations and replace it with a set of discrete equations, called finite-difference equations.
The finite-difference method is typically defined on a regular/structured grid (i.e. the computational domain is usually divided into hexahedral cells (the grid), and the solution will be obtained at each nodal point.) and this fact can be used for very efficient solution methods. Regular grids are useful for very-large-scale simulations on supercomputers.
The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.
Examples include Formula 1 car computational-fluid-dynamic (CFD) simulations and space-shuttle CFD simulations. The finite-difference method is more difficult to use for handling material discontinuities. In addition, it does not lend itself for local grid refinement or anything similar to “adaptive mesh refinement.” This may be needed to resolve local rapid variations in solutions such as around a corner of a complex shape, as described earlier.
But, if we can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are much easier than for finite-volume methods.
If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high-quality results.
The finite-difference method is defined dimension per dimension; this makes it easy to increase the “element order” to get higher-order accuracy.
FVM
The finite-volume method is similar to the finite-differencing method in that the CAD model is first divided into very small but finite-sized elements of geometrically simple shapes. Apart from this, the finite-volume method is very different from the finite-differencing method, beginning with the concept of elements, which are instead referred to as cells.
A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy) and can handle discontinuities in solutions. The PDE is written in a form which can be solved for a given finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the calculation of fluxes is very important in FVM.
The finite-volume method’s strength is that it only needs to do flux evaluation for the cell boundaries. This also holds for nonlinear problems, which makes it extra powerful for robust handling of (nonlinear) conservation laws appearing in transport problems.
The local accuracy of the finite-volume method, such as close to a corner of interest, can be increased by refining the mesh around that corner. However, the functions that approximate the solution when using the finite-volume method cannot be easily made of higher order. This is a disadvantage of the finite-volume method compared to the finite-difference method.
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 in to structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a discrete, but the variables are typically placed at cell centre’s rather than at nodal points. This is not always true, as there are also face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are obtained using interpolation.
FVM can guarantee the local and global conservation of the system.
Interpolation Scheme :
Interpolations schemes contains terms that are interpolations of values typically from cell centres to face centres.
Here w and e are flux .
To find out the flux will use interpolation schemes i.e:
Tw=Tp+TWΔxTw=Tp+TWΔx
Te=TP+TEΔxTe=TP+TEΔx
The interpolationSchemes sub-dictionary contains terms that are interpolations of values typically from cell centres to face centres. being divided into 4 categories: 1 category of general schemes; and, 3 categories of schemes used primarily in conjunction with Gaussian discretisation of convection (divergence) terms in fluid .It is highly unlikely that the user would adopt any of the convection-specific schemes for general field interpolations in the interpolationSchemes sub-dictionary, but, as valid interpolation schemes.
A general scheme is simply specified by quoting the keyword and entry, e.g. a linear scheme is specified as default by:
default linear;
The convection-specific schemes calculate the interpolation based on the flux of the flow velocity. The specification of these schemes requires the name of the flux field on which the interpolation is based; in most OpenFOAM applications this is phi, the name commonly adopted for the surfaceScalarField velocity flux . The 3 categories of convection-specific schemes are referred to in this text as: general convection; normalised variable (NV); and, total variation diminishing (TVD). With the exception of the blended scheme, the general convection and TVD schemes are specified by the scheme and flux, e.g. an upwind scheme based on a flux phi is specified as default by:
default upwind phi;
Some TVD/NVD schemes require a coefficient where corresponds to TVD conformance, usually giving best convergence and corresponds to best accuracy. Running with is generally recommended. A limitedLinear scheme based on a flux phi with is specified as default by:
default limitedLinear phi 1.0;
Apart Interpolation schemes there are something called Flux Limiters.
Flux limiter:
For convective fluid flow, it is ovservedthat low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the vicinity of dinscontinuity or shocks. Highly accurate and oscillation free schemes are known as High Resolution Schemes.
In order to make the scheme TVD, we must constrain or limit the range of possible values of the additional convective flux
which was originally introduced to make the scheme higher-order. Hence, the functionφ(r)φ(r) is called a flux limiter function.
Flux Limiter functions are used to fine tune the higher order and lower order schemes in such a way that the resulting scheme gives a higher order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities. For such a scheme TVD scheme is employed.