Week 9 - FVM Literature Review
Objective :
To describe the need of Interpolation schemes and Flux limiters in FVM (Finite Volume Method).
Finite Volume Method :
The finite volume method (FVM) is technique used in Numerical Analysis to evaluate partial differential equation in the form of Algeabric equation . Similar to the Finite difference method(FDM) or finite element method (FEM), values are calculated at discrete places on a meshed Geometry.
One important feature of the Finite volume schemes is their conservation properties.Since they are based on applying conservation principle over each small control volume , global conservation is also ensured .
" Finite volume " refers to the small volume surrounding each node point on a mesh means the grid points where variables are stored and defined as being in the center of each control volume
The method is initiated by deviding the flow domain into a number of small control volumes.
In the Finite volume method , volume integrals in a partial differential equation that contain a divergence term are converted to surface intergrals.
The advantage of the Finite volume method is that it easily allows to formulate unstructered mesh , and this method is used in many Computational Dynamic packages .
INTERPOLATION SCHEMES IN FINITE VOLUME METHOD :
Interpolation is a process to estimate the values at unknown points by using the points with known value at unknown points by using the points with known value and sample points on either sides of the unknown point . In FVM , Interpolation schemes are used to find values of volume integrals required at the points other than nodes.
The Interpolation schemes are :
- Upwind Interpolation scheme (UDS)
- Linear Interpolation schemes
- Quadratic upwind Interpolation scheme
4 . Hybrid Interpolation scheme
- Total variation Diminishing scheme
- Essentially Non-oscillatory interpolation scheme
- Upwind Interpolation scheme (UDS) :
The Upwind interpolation scheme takes in account the direction of fluid flow and based on it , the value of solution variable at the face is approximated by its value at the upstream computational node.For instance , if the fluid is moving from west to east , then :
`phi_e ~~ phiP`
And if the fluid is moving from east to west (refer figure),then :
`phi_e ~~phiE`
In addition to these Gradient schemes and Interpolation schemes there are flux limiters
Flux limiters
Flux limiters are employed in high resolution schemes, which are numerical systems for solving problems in science and engineering, especially fluid dynamics, that are described by partial differential equations (PDEs). They are employed in high resolution schemes, such as the MUSCL scheme, to eliminate spurious oscillations (wiggles) caused by shocks, discontinuities, or rapid changes in the solution domain that would otherwise occur with high order spatial discretization schemes. The use of flux limiters, together with a suitable high resolution scheme, reduces the total variation of the solutions (TVD).
where
low resolution flux,
high resolution flux,
flux limiter function,
and 
represents the ratio of successive gradients on the solution mesh
This is the limiter case
if limiter function = 0 , then there will be a low resolution and sharp gradient
if limiter function>0 , then there will be a high resolution and +ve gradient.
FDM (Finite Difference Method)
The differential form of the PDE to be solved is used in a finite difference method (FDM) discretization. Each derivative is replaced by a formula that approximates the difference (that can generally be derived from a Taylor series expansion). The solution is obtained at each nodal point, and the computational domain is commonly divided into hexahedral cells (the grid). The FDM is simplest to grasp when the physical grid is Cartesian, but it can be extended to domains that are not simply represented by brick-shaped pieces by using curvilinear transforms.
Finite Volume method (FVM)
The discretization of a finite volume method (FVM) is based on the integral form of the PDE to be solved (e.g. conservation of mass, momentum, or energy). For a given finite volume, the PDE is stated in a form that can be solved (or cell). The computing domain is discretized into finite volumes, and the governing equations are solved for each volume. Fluxes of the conserved variable are frequently included in the resulting system of equations, hence flux calculation is crucial in FVM. The main advantage of this technology over FDM is that it eliminates the need for structured grids and the time and effort required to turn a given mesh into a structured numerical grid.
Finite Element Method (FEM)
A piecewise representation of the solution in terms of defined basis functions is used in a finite element method (FEM) discretization. The computational domain is partitioned into smaller domains (finite elements), and the solution is built from the basis functions in each element. The actual equations are usually found by restating the conservation equation in weak form: the field variables are stated in terms of the basis functions, the equation is multiplied by appropriate test functions, and the equation is then integrated over an element. Because the FEM solution is expressed in terms of specific basis functions, far more information about it is available than for either FDM or FVM.
Difference
Comparison of the three approaches is challenging, owing to the significant differences between them. Discrete solutions are provided by FVM and FDM, while continuous (up to a point) solutions are provided by FEM. FVM and FDM are typically thought to be easier to programme than FEM, however opinions differ. FVM are thought to have higher conservation properties, though opinions differ on this as well.