Week 9 - FVM Literature Review
Finite Volume Method Literature Review
Aim: To describe the Finite volume method (FVM) and its applictions.
Objective:
1. Finite Volume Method(FVM), and the major differences between FDM & FVM.
2. Need for interpolation schemes and flux limiters in FVM.
Solution:
Finite volume method:
In this, entire domain is divided into a finite number of cells, with small volumes.
Cell centres act as nodes and store volume averaged of all variables.
It is based on Guass divergence theorem, which states that integral of the divergence of the given vector over closed volume is equal to the flux of this vector across clsoed surface bonding the volume.
Difference between the FDM and FVM
| FDM | FVM |
| Governing equations are in differential form | Governing equations are in integral form |
| Solution is computed at each node | Solution is computed at cell centroids |
| Requires structured grid to compute | Can handle unstructured grid |
| Suitable only for continuous and differentiable flows | Suitable for discontinuous flow |
| Conservation is not assured | Ensures conservation |
Interpolatoin schemes and flux limiters in FVM:
Interpolation schemes are used to evaluate the faces values that rise apart as convective flux and diffusive flux terms in discretization of Navier stokes equation in integral form.
1. Upwind Interpolation scheme.
It is first order accurate interpolation scheme, bounded and has diffusive character.
Face value is considered as cell centroid of the neighbouring cell in upwind direction.
2. Linear interpolation or central difference scheme.
It is second order accurate interpolation scheme, unbounded and might generate oscillatory solutions.
Face values are calculated from neighbour or owner cells.
3. Linear upwind differencing.
It is second order accurate interpolation scheme, unbounded.
It is used mainly for strong gradient flows.
4. Quadric upwind interpolation scheme.
It is third order accurate interpolation scheme, high oscillatory
Face values are approximated by three-point quadratic function.with two surrounding nodes and one upstream node.
Flux limiters:
These are used for second and higher order interpolations. The values are simulated using cell centroids.
Adding a flux limiter on higher order schemes to first order schemes prevent the formation of new local maxima/minima, these are called as Total variation diminishing scheme.
Flux limiters are also called as slope limiters, because they have same mathematical form.