Importance of Interpolation scheme & Flux Limiter in Finite volume Method

Title:  FVM Literature review

Objective:

1. To describe the need for interpolation schemes 

2. To describe  flux limiters in FVM.

Finite volume Method

     The finite volume method (FVM) is a method for representing and evaluating partial diffrential equation in the form of algebraic equations. This method is apply for over the entire volume & properties are assumed to be concentrated at geometric centre of the volume concerned. Actually finite volume method there is use intergral conservative equation & its evaluated by using function value at compuational node.

Step to solve FVM problem

1. Find out integral form of equation
2. Apply gauss divergence theorem
3. Come up with suitable method to calculate flux term.

Type of scheme

1. Gradient scheme
2. Interpolation scheme

1. Gradient scheme

In gradient scheme for finding the temperature at east face e, then we take the information neighboring point E & this is called gradient scheme.

1. Interpolation scheme

The approximation of surface & volume integral require values of the variable at location other than the computational node of control volume in that case there is use of interpolation scheme.

W= West side volume, P=Present volume, E= East volume

e= east face, w=west face

Example of One-Dimensional heat conduction equatiion

`δ/(δx)(α(δT)/(δx))+S=0`

integral form of governing equation 

`∫ (δ/(δx)(α(δT)/(δx))+S)dv=0`

following is equation of volume integration & surface integral for P volume

`∫_w^eδ/(δx)(α(δT)/(δx))dv+∫Sdv=0`

where dv=A.dx

`∫_w^eδ/(δx)(α(δT)/(δx))A.dx+∫Sdv=0`

inetegrate  with control volume limit

`[(αA(δT]/(δx))]_e^w+sdv=0`

`[(αA(δT]/(δx))]_e-[(αA(δT]/(δx))]_w+barS dv=0`

use of taylor series expansion 

`[(αA(T_E-T_P))/dx]-[(αA(T_P-T_W))/dx]+barS dv=0`

heat flux- heat flux in +source term =0 

above equation called conservative

Type of interpolation scheme

1. Upwind scheme
2. Linear interpolation
3. Quadratic upwind interpolation
4. Hybrid scheme
5. Total variation Diminish (TVD)

1. Upwind scheme(UDS)

Generally upwind scheme are used for convection domain problem. The upwind interpolation for approximate the value of a variable function at the each face of a control volume is given. The UDS is equivalent to using a backward or forward finite difference approximation.

                   Fe=   Fp   if (V.n)e>0      &         Fe   if  (V.n)e<0

If velocity point in positive x-direction , then Fe= Fp

If velocity point in positive x-direction , then Fe= F_E

 Advantage:

1. UDS is the only approximation that satisfies boundedness criterion unconditionally it means that UDS scheme will never yield at oscillatory.

Disadvantage:

1. It is identified that numerical diffusion if multidimensional problem flow is oblique (unstructured) to the grid.
2. Peak or rapid variation in the variable will be smeared out.
3. Only first-order out accurate. Hence very fine grids are required to obtain accurate solution.

2. Linear interpolation(CDS)

Approximate the value of variable at Control volume face center by linear interpolation of the values at two nearest computational nodes. The linear interpolation is equivalent to use of central difference formula of the first order derivative & hence it’s called as central difference scheme.

`Fe=F_E λe+ F_p(1- λe)`

`λe= (Xe-Xp)/( X_E-X_p)`

Advantage

1. Simplest & widely used interpolation
2. Leads to a very simple approximation for evaluation of gradients required for a evaluation of a diffusive flux.
3. Approximation of gradient has second order uniform grids
4. It’s formally first order on non-uniform grids

Disadvantage:

      It may produce oscillatory solution

1. Quadratic upwind interpolation

Approximate the value of variable at Control volume face center by quadratic interpolation of the values at three nearest computational nodes (one downstream node & D two upstream nodes U, UU).

`Fe =Fu+g1* (F_D-F_U) +g2* (F_(UU)-F_U)`

Where

`g_1= ((Xe-Xu) (Xe-Xuu))/((XD-Xu) (XD-Xuu))`

`g2= ((Xe-X_U) (Xe-X_D))/((X_(UU)-Xu) (X_(UU)-X_D))`

 IF Velocity at face e (ue)>0

The downstream node D=E

   For upstream node P=U

For upstream upstream node W=UU

IF Velocity at face e (ue)<0

The downstream node D=P

   For upstream node U=E

For upstream upstream node UU=EE

Advantage

1. Its third order accurate on both uniform & non-uniform grids.

Disadvantage:

      It may produce oscillatory solution

1. Hybrid scheme

It’s combination of CDS & UDS scheme based on local peclet number.

`fe= gamma* f_(CDS)+ (1-gamma)*f_(UDS)`

gamma it is depend on the local peclet number

1. Total variation Diminish (TVD)

`Fe= Fp +1/2 ψ(r)*(Fe-Fp)`

  ψ = Local flux limiter

Flux limiter

 In order to capture shock & discontinuity that arises in hyperbolic equation, in physically this equation model the convective fluid flow. So, it is has been observed that the low- order scheme are usually stable but quite disspative in nature around the point of discontinuity or shocks. On other side higher order numerical scheme are unstable in nature & show oscillation in the vicinity of discontinuity, such behavior by analyzing the modified equation of these scheme.

The main objective of flux limiter is to have highly accurate & stable oscillation free scheme, this scheme called high resolution scheme.

 Define the numerical flux function of high resolution conservative scheme.

`F_(J+1/2) =   L_ (J+1/2 )+phi (H _(J+1/2)- L_(J+1/2))`

Where L, H are numerical flux of conservative law order & high order scheme & phi is a function smoothness parameter theta usually defined by

`Theta= (U_j- U_(j-1))/( U_(j+1)- Uj)`

 To define limite function phi (theta) in such a way that at least the following properties satisfies

1. θ should be positive
2. ϕ(1)=1

Conclusion of flux limiter

The idea is setup the numerical flux of high order & low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in smooth region flow & sticks with first order of accuracy in the vicinity of shocks/ discontinuity.