FVM literature review

Finite volume method:

• Finite volume method is based on the approximation of integral form of the conservative equations.
• The domain is divided into a set of control volumes , the governing equations are solved for each of these control volumes.
• FVM is suitable for complex geometries and it can accomodate any type of grids .
• FVM involves discretizing the domain into control volumes and the equations are solved at each control volume.
• Solution of FVM involves the approximation of the volume and surface integrals usimg the approximation techniques.

Need for interpolation schemes:

• FVM solution process involves the approximation of the volume and surface integrals for the control volume.
• The surface and volume integrals must be evaluated by calculating the values at the other control volumes for this purpose interpolstion schemes are required.
• The interpolation schemes are similar to numerical schemes in finite discretiztaion method.

Types of interpolation schemes:

• Upwind interpolation
• Linear interpolation
• Quadratic upwind interpolation
• Hybrid interpolation schemes

Consider an example of 1D linear heat conduction equation

`int alpha grad T dV +int S dV=0`

where `alpha`-thermal conductivity

S-source term -heat generated

Consider the thermal conductivity as constant.

consider a slab of length l

volume of cv =A.dx

W,P,E are the computational nodes

w,e are the faces

Applying the heat equation to the cv

`A ((alpha (partialT))/(partialx))_e -A ((alpha (partialT))/(partialx))_w+bar S.dV`

now we use the central differencing for the derivatives

`A((alpha(T_E-T_P))/dx)-A((alpha(T_P-T_W))/dx)+barS.dV=0`

The above equation is the discretized equation for the control volume ,the values of temperatures at the faces are to be determined so interpolation schemes are used.

Flux limiters:

• Flux limiters are required in numerical schemes to avoid oscillations in the solutions of high resolution schemes.
• Flux limiters limit the spacial derivatives to realistic values , they are not useful in smooth curves.
  
• They limit the oscillations or wiggles in the derivatives due to shock or discontinuities in the domain.
• CHARM,HCUS,HQUICK are some of the common flux limiters .