FVM Literature
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 location can be obtained by using interpolation formulas, Some of the employable methods are
- Upwind Interpolation Scheme (UDS)
- Central Differencing Approximation (CDS)
- Quadratic Upwind Differencing Scheme (QUICK)
- Hybrid Interpolation Scheme (CDS & UDS)
- Total variation Diminishing Scheme (TVD)
1. UDS :
As the name indicates to the assumption that 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). Mathematically,
`f_e = {(f_p,if,(v.n)_e,>,0),(f_E,if,(v.n)_e,<,0):}`
The values for this at the interface is given by
- For `v>0 ; u_(i-1/2)approxu_(i-1);u_(i+1/2)approxu_i`
`I_c approxv(u_i-u_(i-1))/(Deltax)` - Backward differencing scheme
- For `v>0;u_(i-1/2)approxu_(i-1);u_(i+1/2)approxu_i`
`I_c approx v(u_(i+1)-u_i)/(Deltax)` - Forward differencing scheme
The resulting Taylor series expansion is
`v.u_(i+1/2) = v.u_i - v(Deltax)/2.((delu)/(delx))_(i+1/2)` where the second term is the leading truncation error which resembles a diffusive Flux term.
Upwind schemes are the approximations that satisfies the boundedness unconditionally which turns out to be First order accurate in space.
2. 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_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 firs order derivative and hence this scheme is aslo termed as Central Differencing Scheme(CDS).
Interpolation Polynomial:
`p1(x) = u_L.(x_R-x)/(x_r+x) + 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 approxv.(u_(i+1)-u_(i-1))/(2Deltax)` - Central differencing scheme
Taylor Series Approximation Polynomial:
`u_(i+1/2) = (u_i+u_(i+1))/2 - (Deltax)^2/8.((del^2u)/(delx^2))_(i+1/2) + ...`
The approximation is second order accurate. It is the simplest and most widely used interpolation for evaluation of gradients required for evaluation of diffusvive fluxes.
3. QUICK:
QUICK is a third order Flux approximation. Although it has second order accuract, it approximates the values of the variable at the CV face centre by Quadratic Interpolation of the values at three nearest computational nodes.
`f_e = f_E + g1(f_D-f_U) + g2(f_(UU) - f_U)`
where g1 and g2 is given by
`g1 = ((x_e-x_U)(x_e-x_(UU)))/((x_(UU)-x_U)(x_(UU)-x_D)`
`g2 = ((x_e-x_U)(x_e-x_D))/((x_(UU)-x_U)(x_(UU)-x_D)`
Other Interpolation schemes :
1. Hybrid Interpolation schemes: ( CDS& UDS)
2. Total Variation Diminishing: (TVD)
FLUX LIMITERS:
For convective fluid flow, it is ovservedthat low order schemes are usually stable but quite dissipative in nature around the points of discontinuity or shocks while the higher order schemes are unstable in nature and show oscillations in the vicinity of dinscontinuity or shocks. Highly accurate and oscillation free schemes are known as High Resolution Schemes.
Flux Limiter functions are used to fine tune the higher order and lower order schemes in such a way that the resulting scheme gives a higher 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 scheme is employed.
`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 function where, `r = (u_i-u_(i-1))/(u_(i+1)-u_i)`