Week 9 - FVM Literature Review
Introduction
Finite volume method (FVM) is a discredizing scheme to solve partial differential equations.
Difference between FVM and FDM
1.In finite difference method complex geometries cannot be modeled easily as in finite differnece method the mesh lines generated must be parallel to the x and y coordinate system.The mesh lines in a complex geometry will not be parallel to the cartesian grids. Hence during numerical differentiation an error might be induced in the solution.
2.In the FDM the meshes generated must be in a defined shape,whereas in the FVM the choice of control volume is aribtrary hence complex domain can be modeled easily.
3. In FDM ,the conservation equation is satisfied in the grid points only, whereas in the FVM scheme the conservation equation is satisfied in the entire CV.
4.The FDM uses the differentiation form of the governing equation whereas the FVM scheme uses the integral form of the governing equation.
5.The equation obtained after differentiaation can be physically understood during FVM as compared to FDM scheme.
Why do we need interpoplation schemes?
The integral form of governing equation has surface and volume integrals whose values are unknown.Hence for the approximation of surface and volume integrals we need the values other than those in the computational nodes,hence we use interpolation schemes.
For example:Consider a Contol Volume as shown below
We are integrating the one dimensional heat conduction equation
`del/(delx)[alpha(delT)/(delx)]+S=0`
in this CV.
Consider a small volume `dV`
integrating the heat conduction equation throughout this entire control volume gives us
`int_w^edel/(delx)[alpha(delT)/(delx)]dV+int SdV=0`
dV is nothing but Adx. Therefore
`int_w^edel/(delx)[alpha(delT)/(delx)]Adx+int SAdx=0`
Further simplifying we get
`Aint_e^walpha(delT)/(delx)+int_e^wSdV=0`
The first part can be simplified as
`Aalpha_e[(T_e-T_p)/dx]-Aalpha_w[(T_p-T_w)/dx]`
Now we can futher substitute the computational node values in the equation and obtain the solution.But a problem arises the value of alpha at the surfaces e and w are a function of Temperature at that surfaces,which is unknown. For this reason we use interpolation to obtain temperaturew at unknoewn values using values from known computational nodes.
Linearization of sorce term
`Aalpha_e[(T_e-T_p)/dx]-Aalpha_w[(T_p-T_w)/dx]+int_e^wSdV=0`
In the above equation the source term can be written as an average value `barS`
the integration of which is equal to Su+SpTp where Su and Sp are functions of some arbitary temperature values.
Linearization is done because the source term which is a function of temperation can be other than a linear equation whereas the rest of the term is a linear equation. Hence to standardize the entire equation into being linear the source term is converted into a linear eqaution.
Why flux limiters?
Flux limiters are used to bound the values of gradients. In cases like shock flow problems the discontinuity between the gradient values can be large. Therefore in order to keep the values meaningful and physically realisable we employ flux limiters.