Week 9 - FVM Literature Review

Finite volume method:

it is a method to represent partial differential equations in algebraic form by using a gauss divergence theorem. first, we will consider a nodal point which is enclosed by a finite volume (means we know the volume of that cell) which we will call a cell. volume integrals in the partial differential equation contain a divergence term which is converted to surface integrals using divergence theorem and this term is considered as fluxes and these flux terms are considered that is flux entering a volume is equal to flux leaving the volume so the quantity is conserved.

we can say that in FDM we will solve the equation in a nodal point but in FVM we will solve the equation in a finite control volume 

• instead of writing the equation in a single nodal point, we write it for an entire volume 
• then we apply some mathematical techniques to help us solve this equation 
• also in FVM uses the concept of conservation

steps in FV discretization :

• write the governing equation in its integral form
• apply the gauss divergence theorem
• come up with suitable methods to calculate flux terms

Difference between FDM and FVM :

• in FVM we solve a control volume instead of a nodal point
• even after discretization we can say that flux is conserved in FVM but FDM doesn't.

need of interpolation scheme :

in FVM we need to find the variables at a location other than nodes so it becomes necessary to solve the problem by the interpolation scheme 

for example, let's consider the 1d linear heat conduction equation and after applying FVM and discretization we get`alphaA(T_E -T_P)/(/_\X )-alphaA(T_P-T_W)/(/_\x) +barSdV=0` note that alpha may or may not be the function of temperature. so we can say that to estimate the value of `alpha`we need to find the value of temperature at that face for that we need some interpolation scheme 

there are many interpolation schemes but the most popular are linear or central differencing scheme, upwind  differencing, downwind differencing

for example, let's consider central differencing we know the distance between the central nodal point and its neighboring points and the distance between those two neighboring points length of one of the neighboring points to central point/distance between the neighboring points `psi=(X_P-X_W)/(X_W-X_E)`

we can determine the temperate at point P by 

`T_P=psiT_E+(1-psi)T_W`

like this, we can use some interpolation schemes to calculate the variables in nodal points

flux limiters :

it is observed that for the convective flow problem the interpolation schemes were good but for shock flow problem we can observe some discontinuity in that case we need to use some flux limiters to make sure that this values are close to the real values