Week 9 - FVM Literature Review
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes.
Interpolation schemes
One of the important things in finite volume method is the use of interpolation schemes. These schemes are essentially used to transform the cell-centre quantities to face-centre quantities.
To understand the use of interpolation scheme, consider the example of 1D linear heat conduction equation.
`d/dx(k(dT)/dx) +S=0`
where k is the thermal conductivity and S is the source term for heat generation.
The domain for finite volume method to be applied to the above equation is divided into the following control volumes.
Here the points which are circled are the grid points or cell centres, with point 1 and 8 as boundaries, while the control volume comprises of the volume in between the neighbouring points with cell centres in the middle.
Consider a control volume as shown, with P as the centre point, W and E as boundary points, w and e as the face centres.
Now proceeding with integrating for the given control volume we have,
`int_w^ed/dx(k(dT)/dx)+int_w^eSdx=0`
`[k(dT)/dx]_e-[k(dT)/dx]_w+SDeltax=0` Assuming S as constant
So now we must have values of `(dT)/dx` at 'w' and 'e'. This is where interpolation function comes into picture. We must assume T as some function of x to get the values at 'w' and 'e'. The function must be non-constant else `(dT)/dx` will be zero. Also the function must be continous at those points because at a given face, the heat flux must be continuous.
So, for the present case, a piece-wise linear function between the grid points as shown is chosen. Though the function is discontinuous at gird points, it does not matter as long as it remains continuous at faces.
So the discretised equation finally becomes,
`K_e[(T_e-T_p)/(deltax_e)]-K_w[(T_p-T_w)/(deltax_w)]+SDeltax=0`
It can be written as,
`a_pT_p(i)=a_ET_E(i+1) +a_wT_w(i-1)+b`
where b= `SDeltax`
`a_E=K_e/(deltax_e)`
`a_w=K_w/(deltax_w)`
and `a_p=a_e+a_w`
Thus the above equation can then be solved by using computer program. The interpolation scheme helped to interpolate the values between the cell centres to find `(dT)/dx` at face centres.
Flux Limiters
Flux limiters are used in numerical schemes to solve problems in science and engineering, particularly fluid dynamics, that are described by partial differential equations (PDEs). Their main purpose is to avoid the spurious oscillations (wiggles) that would otherwise occur with high-order spatial discretization(due to shocks, discontinuities, or steep gradients in the solution domain). In simple term, flux limiters as the name suggests are used to limit the spatial derivative values to those which are actually realisable meaningful and avoid the solution from giving unrealistic values.
Some common forms of flux limiter functions are, CHARM, HCUS, HQUICK etc. Flux limiters are also called as slope limiters as there function is to limit the derivative or "slope" terms.