Week 9 - FVM Literature Review

Aim:- To discuss different types of Interpolation schemes and Flux Limiters used in
finite volume method.
Interpolation Schemes in Finite Volume Method:-
1. Upwind Interpolation Scheme(UDS)
2. Central Differencing Schemes(CDS)
3. Quadratic Upwind Differencing Scheme(QUICK)
4. Hybrid Interpolation Scheme(HIS)
5. Total Variation Diminishing scheme(TVD)

Below are the some Interpolation Scheme which explain in detail:-

1. Upwind Interpolation Scheme(UDS):-
In computational physics, upwind schemes basically a numerical discretization method
that use to solve hyperbolic partial differential equations. Upwind schemes use an
adaptive or solution-sensitive finite difference stencil to numerically simulate the
direction of propagation of information in a flow field.

For Backward differencing,

For Forward differencing,

which can written as follows,
`a^+=max(a,0)`, `aoverset(-)()=min(a,0)`

The above two equations can be combined and written as

The above equation is a general way of writing any upwind-type schemes.

2. Quadratic Upwind Differencing Scheme(QUICK):-

In computational Fluid dynamics QUICK, which stands for Quadratic Upstream
Interpolation for Convective Kinematics is a higher-order differencing scheme that
considers a three-point upstream weighted quadratic interpolation for the cell face
values this method can be used to solve the steady convection-diffusion equation.

For this purpose Total Variation Diminishing scheme is used which can be written as
follows:-