Week 9 - FVM Literature Review

There arer three main methods in cfd to solve differential equations, they are:

Finite element method (FEM),

Finite difference method(FVM) and,

Finite volume method(FVM).

Finite volume methods are mainly used for solving convection and diffusion problems, consiting partial difference equations. Firstly, we convert the partial differential equations into algebraic froms, and the solutions are calculated for the finite or discrete volumes. The dependent variable's values for the governing equations are stored at the centre of the given volume.

The finite volume method is a conservative method because in this method, the flux which enters the volume is considered to be equal to the flux that leaves the volume. The entire volume is dicretised into smaller control volumes and can be used on complex and unstructured mesh.

On the other hand, finite difference method, nodes are the place where the values for the dependent variable are stored. All the grids are linear with the grid points arranged in a straight line. Without proper alignment of the grid points, finite difference method cannot be used and the correspodning parameters cannod be calculated therefore, leading us to find an approximation method to obtain the results. Also, unlike finite volume method, finite difference method can only be used for less complex problems.

Interpolation in finite volume method is the process of estimation of values at unknown points by using the values at known points i.e. if we know that values for the preceeding and succeeding points for a given unknown point, we use these values to calculate the value for tehe unknown point.

There are different interpolation schemes used in finite volume method. they are:

Upwind interpolation scheme: used in convection dominated problems.This method is accurate for first order problems. For the approximation of a value at the east face of a given control volume:

`f_e = f _p if ( v . n ) e > 0 and f_e=f _E if ( v . n ) e < 0`

for v>0, the values at the interface are given as:

`u_i − 1/2 ≈ u_(i − 1) , u_i + 1/2 ≈ u i`

and the backward difference scheme: `I c ≈ v (u_i − u_(i − 1) )/(Δ x)`

for v<0:

`u _i − 1 /2 ≈ u _i , u _i + 1 /2 ≈ u_ (i + 1`

and the forward difference scheme: `I c ≈ v( u_( i + 1) − u _i)/( Δ x)`

`v u_( i + 1 /2) = (v u _i) − (v Δ x )/2 ( (∂ u)/ (∂ x) )_ (i + 1 /2) − (v ( Δ x )^ 2 /8) (( ∂ ^2 u )/(∂ x ^2 )) _(i + 1 /2) + ...` is the taylor series expansion.

Linear interpolation scheme: this method is usefulfor approximating the values for a given control volume at the centre of the face by linear interpolation of values at the nearest computational nodes similar to the central differencing scheme. This method is accurate for second order problems.

`f _e = f_ E λ _e + f_ p ( 1 − λ _e ) ; λ_ e = (x _e − x _p)/( x _E − x _p)`

values at interface:

`u _(i − 1 /2) ≈ (u _(i − 1) + u _i)/ 2 and u _(i + 1 /2) ≈ (u_ i + u _(i + 1) )/2`

`I _c ≈ v (u _(i + 1) − u_( i − 1))/( 2 . Δ x)`

after finding the taylor series approximation for u(i) and u(i+1), we get:

`u_( i + 1 /2) = (u _i + u _(i + 1 ))/2 − (( Δ x ) ^2)/ 8 . ( (∂ ^2 u) /(∂ x^ 2) ) _(i + 1 /2 )+ ...`

Quadratic interpolation scheme: in this method, the values at the nearest 3 points are approximated using quadratic interpolation for the values of variables given at the center of a given control volume face. This method is accurate for third order and fourth order problems.

`f _e = f _U + g _1 ( f _D − f _U ) + g _2 ( f _(U U) − f_ u )`

`g _1 = (( x _e − x _U ) *( x _e − x _(U U) ))/ (( x _D − x _U ) ( x _D − x _(U U)) )`

`g _2 = (( x _e − x _U ) *( x _e − x _(D) ))/ (( x _(UU) − x _D ) ( x _(UU) − x _(U)) )`

Therefore, from the above expressions, the relation obtaine dis as follows:

`ϕ _e = (6 /8) ϕ _c + (3 /8) ϕ _D − (1/ 8) ϕ _(U U)`

Hybrid interpolation scheme: This method is based upon the peclet number and is a combination of upwind differencing scheme and central differencing scheme.

`f _e = γ f _(C D ) + ( 1 − γ ) *f _(U D )`

Total variation diminishing scheme: it is a method used to solve hyperbolic pde's of higher orders and is based upon discretisation schems.

`f _e = f _p + 1/ 2 ψ ( r ) ( f _E − f _p )`

Flux limiters: Flux limiters are used to obtain higher order accuracy and lower order accuracies depending upon the condition of flow. Flux limiters are used in high resolution schmes such as total variation diminishing schemes to solve partial differential equations.