Week 9 - FVM Literature Review

Study of various interpolation schemes and flux limiters in FVM. 

Finite volume method :

The finite volume method (FVM) is a method for representing and evaluating partial diffrential 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 theorm. 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. The method is used in many computational fluid dynamics packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.

Interpolation Schemes in Finite Volume Method (FVM):

The interpolation schemes used in FVM method are as follows

1. Upwind Interpolation Scheme. (UDS)
2. Central Differencing Approximation. (CDS)
3. Quadratic Upwind Differencing Scheme. (QUICK)
4. Hybrid Interpolation Scheme. (CDS & UDS)
5. Total variation Diminishing Scheme. (TVD)
   
   

1. Upwind Interpolation Scheme(UDS):

In this scheme we use the node in the upwind or upstream of the given node. It is generally used for convection dominated problems (i.e Pe>1). ,

$f_e=\{\begin{array}{cccc}{f}_p,if& {(v.n)}_e& >& 0\\ f_E,if& {(v.n)}_e& <& 0\end{array}$

The values for this at the interface is given by

For $v>0;u_{i-\frac{1}{2}}\approx u_{i-1};u_{i+\frac{1}{2}}\approx u_i$
$I_c\approx v\frac{u_i-u_{i-1}}{\Delta x}$ - (Backward differencing scheme)

For  $v>0;u_{i-\frac{1}{2}}\approx u_{i-1};u_{i+\frac{1}{2}}\approx u_i$
$I_c\approx v\frac{u_{i+1}-u_i}{\Delta x}$ - (Forward differencing scheme)

 Taylor series expansion is 

$v.u_{i+\frac{1}{2}}=v.u_i-\left(v\frac{\Delta x}{2}\right).{\left(\frac{\partial \left(u\right)}{\partial \left(x\right)}\right)}_{i+\frac{1}{2}}$  where the second term has the truncation error which resembles a diffusive Flux term.

Upwind schemes are the approximations that satisfy the boundedness unconditionally and is First order accurate in space.

2.Central Differencing Approximation (CDS):

Approximate values of the variable at the control volume (CV) face centre by the linear interpolation of the values at the two nearest computational nodes.

$f_e=f_E{\lambda }_e+f_p(1-{\lambda }_e)$  where  ${\lambda }_e=\frac{x_e-x_p}{x_E-x_P}$

The linear interpolation is equivalent to that of using central difference formula of the first order derivative and hence this scheme is also termed as Central Differencing Scheme(CDS).

3. Interpolation Polynomial:

$p1\left(x\right)=u_L.\frac{x_R-x}{x_r+x}+u_R.\frac{x-x_L}{x_R-x_L}$

The interface values are given by:

$u_{i-\frac{1}{2}}\approx \frac{u_{i-1}+u_i}{2}$ and $u_{i+\frac{1}{2}}\approx \frac{u_i+u_{i+1}}{2}$

$I_c\approx v.\frac{u_{i+1}-u_{i-1}}{2\Delta x}$ - (Central differencing scheme)

Taylor Series Approximation Polynomial:

$u_{i+\frac{1}{2}}=\frac{u_i+u_{i+1}}{2}-\left(\frac{{(\Delta x)}^2}{8}\right).{\left(\frac{{\partial }^2\left(u\right)}{\partial {\left(x\right)}^2}\right)}_{i+\frac{1}{2}}+...$

The approximation is second order precise. It is the simplest and most widely used interpolation technique for the evaluation of gradients required to measure diffusive fluxes.

FLUX LIMITERS:

Flux limiters are used in the numerical schemes to solve problems of fluid dynamics, described by highly coupled non-linear Partial Differential Equations. If we tend to use low order numerical schemes for the solving of the governing equations then we might get a highly oscillatory and unstable solution near the discontinuity if present. Whereas if we tend to use higher-order numerical schemes to capture the phenomenon more accurately we tend to exploit the computational power unnecessarily. The main idea behind the use of the flux limiters is to limit the spatial derivative terms near the discontinuities to physical real values so as to maintain physical consistency. There are several types of flux/slope limiter functions. They come in handy when sharp wave-fronts/peaks are present. For smoothly changing waves/functions, the flux limiters do not operate and the spatial derivatives can be represented normally.

It is imperative to note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities.

$F\left(u_{i+\frac{1}{2}}\right)=f^{low}(i+\frac{1}{2})-\varphi \left(r_i\right)(f^{low}(i+\frac{1}{2})-f^{high}(i+\frac{1}{2}))$

$F\left(u_{i-\frac{1}{2}}\right)=f^{low}(i-\frac{1}{2})-\varphi (r_i-1)(f^{low}(i-\frac{1}{2})-f^{high}(i-\frac{1}{2}))$

Here$f^{low}$ = Low precision flux (1st order precise)

          $f^{high}$= High precision flux (Highest order precision)

           $\varphi \left(r\right)$ = Flux limiter function and $r=\frac{u_i-u_{i-1}}{u_{i+1}-u_i}$.