Week 9 - FVM Literature Review

Interpolation schemes:

Interpolation schemes are used to transfer cell-centre quantities to the face centres that means the Interpolation schemes sub-dictionary contains terms that are interpolations of values typically from cell centres to face centres, primarily used in the interpolation of the velocity to face centres for the calculation of flux phi.

Flux Limiters:

Flux limiters are used to control the range of values to which it can jump to avoid the occurence of Spurious Oscillations that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain.

Now, let us consider a case of 1D linear heat conduction equation which has the following governing differential equation:

where, alpha is the thermal conductivity and S is the source term.

Now, the integral form of the equation is:

Now further integrating it and applying the boundary conditions, we have the equation as follows:

where the first two terms are the heat flux and the S(bar) term is the average of the source over the differential volume which needs to be linearized beacuse while approximating the particular quantity by FVM, we need to account for all the non-linearities in the equation which gives the mechanism to iteratively approximate the non-linearity and thereafter convert it into linear form for better approximation. After linearizing using Picard's Method, we have the following equation:

where 'Sc' is constant and 'SpTp' is the iterative part.