Week 9 - FVM Literature Review
FINITE VOLUME METHOD
It is the method of converting partial differential equations into algebraic form of equations in order to solve them. Then these equations are solved by using iterative solvers or matrix inversion method.
Gauss Divergence theorem is used to convert volume integral from of governing equation into surface integral .
These equations are solved for flux terms. The flux entering through control volume is equal to flux leaving the control volume. Hence, the resulting solution is always coserved which is not true in case of finite difference method.
Finite difference method is only useful when the mesh is of structured form, while Finite volume method can be applied to unstructured mesh as well.
The major differences between finite volume method and finite difference method are as below,
1. Since the governing equations are conservative in nature, FVM solves governing PDEs over total control volume using the gauss divergence theorem which also results in conservative form.
2. One of the major advantage of FVM is that it only needs to calculate the flux variables at the volume boundaries. This can be applied to all non-linear conservative governing equations. This is very important and also the reason why it is widely used in CFD applications.
3. The accuracy of the FVM can be increased by making the mesh finer in that region.
4. It is difficult to obtain higher- order schemes in the finite volume method when compared to finite difference method.
5. The accuracy and stability of the solution depends upon initial and boundary conditions in FDM.
Most of flow problems in CFD consist of convection and diffusion terms in which linear interpolation scheme that is CDS can not be used as it may give errors as it may give errors because of large oscillations if finer mesh not used. Due to this, there was a need to use other interpolation schemes. Irrespective of differencing methods used numerical results obtained are close to the exact solution of the governing equation when the number of cells in large but are results of the discretisation scheme will be physically realistic when it has certain fundamental properties, which are
1) Coservativeness : The integration of the convection- diffusion equation over a finite number of control volumes result in discretized conservative equations involving fluxes. For the equations to be conserved the flux leaving the control volume face should be equal to flux entering the control volume face.
2) Boundedness: This property means that an increase in the flow variable at one node willl result in an increase of the flow variables in neighbouring nodes. If the discretization scheme does not satisfy the condition of boundedness, then the solution does not converge it contains wiggles.
The conditions for boundedness are :
a) All coefficients of the discretized equations should have the same sign.
b) The resulting matrix from the discretized should be diagonally dominant.
3) Transportiveness : It defines the relationship between directionality of the influencing and the flow direction and magnitude of the Peclet number.
Flux limiters :
We have seen the behaviour of numeric schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuitie/shock. On other hand, higher order numerical schemes are unstable in nature and show oscillations in viscnity of discontinuity.
However, we can not have higher order accuracy without oscillations. The objective is to hahve accurate and stable oscillation free schemes . This kind of schmes are known as high resolution schemes. To obtain such a scheme, the idea is to tune the numerical flux of high order and low order schemes using the flux limiter function in such a way that the resulting schemes gives a higher order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of discontinuity.
They operate functionally as a switch and tune the numerical fluxes obtained from lower order scheme such that near the discontinuity lower order scheme is employed.
Use of flux limiter is to prevent the solution from oscillating and to preserve the accuracy.
