Finite Difference Method (FDM) V/s Finite Volume Method (FVM) Literature Review

Finite Difference Method (FDM) V/s Finite Volume Method (FVM)

In Computational Fluid Dynamics (CFD) the governing equations defining the physics i.e., navier stokes equations are solved numerically. These equations are in partial differential form which can’t be solved directly. So we employ a process called discretization.

Discretization – Conversion of partial differential equation (PDE) form to algebraic form. This can be done by three different techniques,

1. FDM
2. FVM
3. FEM

Finite Difference Method (FDM)

1. It is the most direct approach to convert the governing equations to algebraic form. In this method, a node in a space is considered for solving the discrete equations and is solved at every node.
2. It uses the information only from the grid points for approximation of any function.
3. FDM is used mainly for structured grids i.e., proper block like shaped domains.
4. Regular grids can be used in some domains such as Meteorological, seismological and astrophysical simulations.
5. It is very difficult to implement FDM in curved boundaries, so for irregular shaped domain, grid variable transformation approach is required to solve.

Finite Volume Method (FVM)

1. It is different from the FDM. In this method, the entire domain is divided into small but finite volumes called cells.
2. It uses the averages from the cell/finite volumes for approximation of any function.
3. It is based on the conservation principle. The idea is that anything coming into the cell from one boundary needs to leave the cell from the other boundary i.e., flux evaluation.
4. The governing equations are solved at every cell volume, where centroid of each cell represents the cells and considered to have the averaged quantities of all flow variables.
5. FVM can be used for unstructured grid as well. This method is very much preferred and suitable for fluid flow simulations, heat transfer, etc.

Need for Interpolation schemes

1. In Finite volume method (FVM), the flow variables are stored in the centroid (P, N) of the cell as shown in the figure below.

1. The navier stokes equation is integrated across the cell volume i.e., volume integrals are solved for each cell. The navier stokes equation normally contains convection term, pressure gradient, diffusion term and a source term.
2. For solving the convection & diffusion terms, the divergence theorem is used and the volume integrals are converted into surface integrals.
3. Each cell can have numbers of faces depending on the type of mesh and each face will be associated with a neighbouring cell. The velocities at centre of each face of the corresponding cell are computed.
4. In order to compute the face centre velocity the interpolation in needed as the face centre velocity is calculated using velocity at the owner and neighbour cell centroids.
5. In setting up a CFD code - discretization is a very basic and while choosing the discretization schemes we are actually choosing an interpolation method.
6. There are different interpolation schemes available such as Upwind, Linear upwind, Central Differencing, and QUICK. 
7. These schemes calculate the face centre values using information at the owner and neighbour cell centroids.

Flux Limiters

1. Flux limiter is a mechanism to address the source of oscillation especially the non-physical oscillation in an extrapolation.
2. Extrapolation is good in a smooth region but it’s bad especially near the near discontinuities.
3. As the FVM deals with the cell averages, if the cell average of current cell is greater or less than its left or right neighbouring cell’s average then the sudden oscillation will occur due to the local maxima and minima.
4. So flux limiter detects such situation i.e., the local maxima or minima then turns off the extrapolation from one cell to another and goes back to piece wise constant approximation.
5. But if the cell averages of current and neighbouring cells are almost the same then extrapolation is used.
6. It’s basically a switch as it switches between extrapolation and piece wise constant approximation depending on the situation. So for a smooth switching and continuous solution from one cell to another flux limiters are very crucial.