FVM Literature Review
Interpolation Schemes in Finite Volume Method(FVM):
The approximation of surface and volume integrals may require values of the variable at locations other than the computational nodes of the CV. The values at this locations can be obtained by using interpolation formulas, some of the methods are:
- Upwind Interpolation Schemes: (UDS)
- Central Differencing Approximation: (CDS)
- Quadratic Upwind Differencing Scheme: (QUICK)
- Hybrid Interpolation Schemes: (CDS and UDS)
- Total variation Diminishing Scheme: (TVD)
1. Upwind Interpolation Scheme: (UDS)
As the indicates to the assumption that we use the node in the upwind or upstream of the given node. It is generally used for covenction dominated problems (i.e Pe > 1). It can be represented as follows:
`f_e = { (f_p if (v.n)_e > 0),(f_E if (v.n)_e <0):}`
The values for this interface is given by:
for v>0, we have: `u_(i-1/2) approx u_(i-1);u_(i+1/2) approx u_i`
`I_c approx v (u_i - u_(i-1))/(Delta x)` - Backward Differencing Scheme
for v > 0 we have: `u_(i-1/2) approx u_i; u_(i+1/2) approx u_(i+1)`
`I_c approx v(u_(i+1) - u_i)/(Delta x)` - Forward Differencing Scheme
The resulting Taylor series expansion is `vu_(i+1/2) = vu_i - v (Deltax)/2.((del u)/(delx))_(i+1/2)` where the second term is the leading truncation error which resembles a diffusive flux.
Upwind Differencing scheme is the only approximation that satisfies boundedness creation unconditionally. It is first order accurate in sapce.
2. Central Differencing Approximation(CDS) :
Approximate values of the variable at the Control Volume face centre by linear interpoation of the values at two nearest computaional nodes.
`f_e = f_Elambda_e + f_p(1-lambda_e) where, lambda_e = (x_e - x_P)/(x_E - x_P)`
The linear interpolation is equivalent to the use of central difference formula of the first order derivative, and hence, this scheme is also termed as Central Differencing Scheme(CDS).
Interpolation Polynomial:
`p_1(x) = u_L.(x_R - x)/(x_R - x_L) + u_R.(x-x_L)/(x_R - x_L)`
Average Interface Values are given by,
`u_(i-1/2) approx (u_(i-1) + u_i)/2`and `u_(i+1/2) approx (u_i+u_(i+1))/2`
`I_c approx v.(u_(i+1) - u_(i-1))/(2Deltax)` - Central Differencing
Taylor Series Approximation
`u_(i+1) = u_(i+1/2) + (Deltax)/2.((delu)/(delx))_(i+1/2) + (Deltax)^2/8.((del^2u)/(delx^2))_(i+1/2) + ....`
`u_i = u_(i+1/2) - (Deltax)/2.((delu)/(delx))_(i+1/2) + (Deltax)^2/8.((del^2u)/(delx^2))_(i+1/2) + ....`
From the above expression we get:
`u_(i+1/2) = (u_i+u_(i+1))/2 - (Deltax)^2/8 . ((del^2 u)/(del x^2))_(i+1/2) + ...`
The aprroximation is second order accurate. It is the simplest and most widely used interpolation for evaluation of gradients required for evalution of diffusive fluxes.
3. Quadratic Upwind Differencing Scheme: (QUICK)
QUICK stands for Quadratic Upwind Interpolation scheme for Convective Kinematics. It is a third order flux approximation. Although it has second order overall accuracy, it approximates the values of the variable at CV face centre by Quadratic Interpolation of the values at three nearest compuational nodes.
`f_e = f_E + g_1(f_D - f_U) + g_2(f_(UU) - f_U)`
where `g_1`and `g_2` is given by:
`g_1 = ((x_e - x_U)(x_e - x_(UU)))/((x_(UU) - x_U)(x_(UU)-x_D)`
`g_2 = ((x_e - x_U)(x_e - x_D))/((x_(UU) - x_U)(x_(UU)-x_D)`
Other Interpolation Schemes:
1. Hybrid Interpolation Schemes: (CDS and UDS)
2. Total Variation Diminishing: (TVD scheme)
FLUX LIMITERS:
For convective fluid flow, it is observed that low order schemes are usually stable but quite dissipative in nature around points of discontinuity or shocks while the higher order schems are unstable in nature and show oscillations in the vicinity of discontinuity. Highly accurate and oscillation free schemes are known as High Resoltion Schemes.
Flux limiter functions are used to tune high order and low order schemes in such a way that the resulating scheme gives a high order accuracy in the smooth region of the flow and maintains first order accuracy in the vicinity of shocks and discontinuities. For such a scheme TVD criteria is maintained. The basic forms are shown below:
`F(u_(i+1/2)) = f^(low)(i+1/2) - phi(r_i)(f^(low)(i+1/2) - f^(high)(i+1/2))`
`F(u_(i-1/2)) = f^(low)(i-1/2) - phi(r_(i-1))(f^(low)(i-1/2) - f^(high)(i-1/2))`
where `f^(low)` = low precision flux (1st order accurate)
`f^(high)` = high precision flux (highest order accurate)
`phi(r)` = Flux Limiter fnction where `r = (u_i - u_(i-1))/(u_(i+1) - u_i)`
Some of the the common forms of Flux Limiter Functions: CHARM, HCUS< HQUICK, Koren, minmod etc.,