Literature review on the need for interpolation schemes and flux limiters in Finite Volume Method (FVM)
Aim: A literature review on the need for interpolation schemes and flux limiters in Finite Volume Method (FVM)
Types of Numerical Methods used in CFD
1) Finite Difference Method ( FDM )
2) Finite Element Method ( FEM )
3) Finite Volume Method ( FVM )
In the above figure, P and E are computational nodes of a controlled volume in a Finite Volume Method. e is a point that lies between two computational nodes.
Interpolation Schemes:
To find a solution or variable at point e an interpolation scheme is required. Accuracy of the solution depends on the accuracy and selection of the correct interpolation scheme depending on the type of problem.
Flux limiters:
Flux limiters are specified constants/functions which depend on the nature of the problem and the interpolation scheme used in the problem. Flux limiters tell the program where to start/stop interpolation schemes and limit the solution to avoid unnecessary high diminishing of results and preventing solution high oscillations in solution.
Types of interpolation schemes
Upwind Interpolation scheme:
`f_e = {(f_p if (v.n)_e > 0),(f_E if (v.n)_e < 0):}`
or
`f_E = if (v.n)_e < 0`
lets assume, `v.n > 0`
`f_e ~~ f_p` ... (i)
Using Taylor's expansion
`f(x) ~~ f_p + ( x - x_p ) ((delf)/(delx))_p + ( x - x_p)^2 ((del^2f)/(delx^2))_p + ...` ... ( ii)
Value of function at e
`f_e = f_p + (x_e - x_p ) ((delf)/(delx))_p + (x_e - x_p)^2 ((del^2f)/(delx^2))_p + ...` ...(iii)
let error = `epsilon`
`epsilon = (x_e - x_ p) ((delf)/(delx)) + ...` ... (iv)
error is 1st order accurate
let, diffusive flux be `f_d = rho*u*phi` ...(v`
`epsi = [rho(x_e - x_p) u]((delphi)/(delx))_p` ... (vi)
compare `epsi` in (vi) w.r.t diffusive flux `f_d`
`f_d = Gamma ((delphi)/(delx))`
where, `Gamma`= diffusivity
Hence, error (`epsi`) represents an artificial diffusivity (`Gamma`) which helps to approximate the solution between two computational nodes.
Advantages:
Satisfies Boundedness criteria unconditionally (irrespective to the mesh/grid size)
Helps to damp unnecessary oscillations
Better suitable for convective problems
Disadvantages:
Removes oscillatory results and diminishes them even if the solutions actual pattern is in oscillatory type.
Linear interpolation schemes:
A linear interpolation scheme is also known as the central difference scheme [CDS]
`f_e = f_E lambda_e + f_p (1-lambda_e)`
where,
`lambda_e = (x_e - x_p)/(x_E - x_p)`
error (`epsi`) = `(x_e - x_p)^2`
Advantages:
Second order accurate ( accuracy is equal to central difference scheme )
Suitable for Oscillatory type solutions
Disadvantages
Can produce oscillatory results even in advection problems where damped solution is required
QUICK interpolation schemes [Quadratic Upwind Interpolation for Convective Kinematics)
Uses three nearest computational nodes ( 1 downstream and 2 upstream nodes) to find the solution by approximating the variable at controlled volume center face by quadratic interpolation)
If `u_e > 0`
D = E
U = P
UU = W
If `u_e < 0`
D = P
U = E
UU = EE
`f_e = f_U + g_1 ( f_D - f_U) + g_2(f_(UU) - f_U)`
`g_1 = ((x_e - x_U)(x_e - x_(UU)))/((x_D - x_U)(x_D-x_(UU))`
`g_2 = ((x_e - x_U)(x_e-x_D))/((x_(UU)-x_U)(x_(UU)-x_D)`
Advantage:
Third order accurate
Suitable for convective and quadratic problems
Has better damping as well as oscillation problem accuracy
Hybrid Interpolation scheme
Combination of Central difference scheme and upstream difference scheme based on local peclet number
`f_e = gammaf_(CDS) + (1-gamma)f_(UDS)`
Total Variation Diminishing (TVD)
`f_e = f_p + 1/2 psi (r) (f_E - f_P)`
`psi * (r)` = specified based on problem
`psi =`flux limiter