Week 9 - FVM Literature Review
The Finite Volume Method (FVM)
The Finite Volume Method (FVM) is a technique used in the solution of partial differential equations representing the conservation laws. FVM is a numerical technique that transforms the partial differential equations over differential volumes into discrete algebraic equations over finite volumes. The first step in the solution process is the discretization of the geometric domain. In the FVM the domain is discretized into non-overlapping elements or finite volumes. The partial differential equations are then discretized/transformed into algebraic equations by integrating them over each discrete element. The system of algebraic equations is then solved to compute the values of the dependent variable for each of the elements.
In the finite volume method, some of the terms in the conservation equation are turned into face fluxes and evaluated at the finite volume faces. Because the flux entering a given volume is identical to that leaving the adjacent volume, the FVM is strictly conservative. This inherent conservation property of the FVM makes it the preferred method in CFD. Another important attribute of the FVM is that it can be formulated in the physical space on unstructured polygonal meshes. Finally, in the FVM it is quite easy to implement a variety of boundary conditions in a noninvasive manner since the unknown variables are evaluated at the centroids of the volume elements, not at their boundary faces.
The difference between the Finite Volume Method and the Finite Difference Method(FDM)
A Finite Difference Method is based on discretizing the PDE using the finite difference approximation formulas. The domain of the solution will be divided into hexahedral grids and the solution is calculated at each nodal point. The solution using the FDM will be easy for cartesian elements, but if the geometry is curvilinear or for complex geometries the solution procedure requires the transformation of the complex geometry to a cartesian geometry using an appropriate transfer function which will not be an easy task
A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved. The PDE is written in a form that 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 FVM 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. As with FDM, the resulting approximate solution is 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. FVM is generally expected to provide better conservation properties.
Finite Volume Interpolation Schemes
Interpolation schemes in FVM are used to calculate the approximations of surface and volume integrals at locations other than the nodes of the control volume. Values at these locations are calculated using interpolation formulas.
Upwind Interpolation
The Upwind Interpolation or the Upwind Difference Scheme (UDS) is the first-order scheme. This scheme is used in the case of convection problems. The scheme is also numerically diffusive. The value at any point is approximated to be the value at the east side of the point if the flow direction is from east to west. If the flow is from west to east the value at that point is approximated to be that on the west side. The UDS can be assumed to be a forward difference or a backward difference scheme depending on the direction of flow.
Linear Interpolation
The value of a variable at the face center of any control volume can be calculated using a linear interpolation scheme as shown below
`\begin{equation} f_{\mathrm{e}}=f_{\mathrm{E}} \lambda_{\mathrm{e}}+f_{\mathrm{P}}\left(1-\lambda_{\mathrm{e}}\right), \quad \lambda_{\mathrm{e}}=\frac{x_{\mathrm{e}}-x_{\mathrm{p}}}{x_{\mathrm{E}}-x_{\mathrm{p}}} \end{equation}`
This interpolation scheme is similar to a central difference scheme for first order derivative. The scheme is second order accurate and may produce an oscillatory solution. Where , E, e, P,W w etc depends on the flow directions.
The Quadratic Upwind Interpolation (QUICK)
The QUICK interpolation scheme can be derived using polynomial fitting. The QUICK scheme on a uniform cartesian grid is given by:
`\begin{equation} \phi_{\mathrm{e}}=\frac{6}{8} \phi_{\mathrm{U}}+\frac{3}{8} \phi_{\mathrm{D}}-\frac{1}{8} \phi_{\mathrm{UU}} \end{equation}`
where D, U and UU represents the donstream upstream and the second upstream node respectively. The scheme is third order accurate but the solution is oscillatory.
Total Variation Diminishing (TVD) Scheme
The upwind scheme is of the first order, but its the most stable scheme whereas the other higher-order accurate schemes are oscillatory in nature. The desired property for a stable, non-oscillatory, higher-order scheme has the property that the total variation of the discrete solution should diminish with time.
Flux Limiters
In modelling the convective fluid flow, it has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order, numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. It is important to note that we cant have higher-order schemes without oscillations. The objective is to create higher-order accurate schemes without oscillations known as high-resolution schemes. The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with the first order of accuracy in the vicinity of shocks/discontinuities as follows:
`\begin{equation} F_{j+\frac{1}{2}}=L_{j+\frac{1}{2}}+\phi(\theta)\left(H_{j+\frac{1}{2}}-L_{j+\frac{1}{2}}\right) \end{equation}`
where `L` and `H` are the numerical flux of conservative low order and high order schemes respectively, `\phi`
is a function of smoothness and `\theta` is defined as
`\begin{equation} \theta_{j}=\frac{U_{j}-U_{j-1}}{U_{j+1}-U j} \end{equation}`
the smoothness function `\phi(\theta)` must satisfy the following properties:
`\phi(\theta )` must be positive for all `\theta`
`\phi(1) = 1`
passes through a particular region known as TVD region associated with the underlying scheme in order to guarantee stability of the scheme.