Week 9 - FVM Literature Review
The finite volume method:
FVM can be used on all differential equations, which can be written in the divergence form. This effectively writes the equation using divergence operators. The equation is then integrated over the volume. We can then apply Gauss’s theorem converting the volume integral over the divergence into a surface integral across the boundaries.
The integral is therefore turned from integrating the differential of the dependent variable inside of the cells into surface integrals of the fluxes of the dependent variable across the boundary of the cells. These integrals can usually be calculated using suitable numerical approximation methods. This simplifies the differential equation substantially. We have used similar approaches previously, e.g., when deriving the mass conservation using Gauss’s theorem. This approach was based on the concept that all mass that would “diverge” out of the control volume must inherently pass the boundary of the control volume at some time. If this flux is integrated over time, the total change of mass in the control volume can be derived. FVM uses the same approach. By monitoring the fluxes of the dependent variable across the boundary of the cells, a conservation approach of the dependent variable is obtained. This is in contrast to FDM, which seeks to approximate the differentials of the differential equation, i.e., the changes of the dependent variable across the cell.
FVM solvers range among the most popular methods in CFD and there are numerous commercial software packages that use this method.
Difference between FVM and FDM:
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. 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.
Interpolation schemes in Finite Volume Method:
Interpolation is a process to estimate the values at known points by using the points with known value and sample points on either sides of the unknown point. In FVM the interpolation scheme are used to find values of volume integrals required at the points other than nodes.
The different interpolation schemes are:
1)Upwind interpolation scheme(USD)
2)Linear interpolation scheme
3)Quadratic Upwind interpolation scheme
4)Hybrid interpolation scheme
5)Total variation Diminishing scheme
6)Essentially Non-oscillatory interpolation scheme
Upwind interpolation scheme:
The upward interpolation scheme for approximating the value of a variable 'f' at the east face of a control volume is given by
It is generally used convection dominated problems, the values at the interface is given by
Flux limiters:
Flux limiters are used in high resolution scheme – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDEs). They are used in high resolution schemes, such as the 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).
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. In general, the term flux limiter is used when the limiter acts on system fluxes, and slope limiter is used when the limiter acts on system states (like pressure, velocity etc.).