Week 9 - FVM Literature Review
FVM:
- The basis of the Finite Volume Method is the integral conservation laws. The essential idea is to divide the domain into the many control volumes or cells & approximates the integral conservation law on each of the control volumes.
FDM vs 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 (the grid), and the solution will be obtained at each nodal point. The FDM is easiest to understand when the physical grid is Cartesian, but through the use of curvilinear transforms the method can be extended to domains that are not easily represented by brick-shaped elements. The discretization results in a system of equation of the variable at nodal points, and once a solution is found, then we have a discrete representation of the solution.
- 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). 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 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 centers rather than at nodal points.
Importance of Interpolation Scheme:
- 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. The interpolation schemes used in FVM are as follows:
1. Upwind Interpolation Scheme (UPS)
2. Central Differencing Approximations (CDS)
3. Quadratic Upwind Interpolation Differencing Scheme (QUICK)
4. Hybrid Interpolation Scheme (UPS & CDS)
5. Total Variation Diminishing (TVD) Scheme
Flux Limiters:
For convective fluid flow, it has been observed that low-order schemes are usually stable but quite dissipative in nature around points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the vicinity of discontinuity. Highly accurate & oscillation free schemes are known as high resolution schemes.
Flux limiters basically used in high resolution schemes, to avoid the random & violent oscillations that can occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain which is not really much appreciated. The login behind the ocnstruction of flux limiter schemes is to limit the spatial derivatives to values which are close to the real world datas.
Flux limiter functions are used to tune high order & low order schemes in such a way that the resulting scheme give a high order accuracy in the smooth region of the flow & maintains first order accuracy in the vicinity of shocks & discontinuities.
References:
1. https://www.cfd-online.com/Forums/fluent/39028-difference-between-fem-fdm-fvm.html