Boris and Book (1973) describe a two-stage technique which is conservative, monotone preserving and enforces the non-negative property of the profile, called Flux corrected transport.
The first stage involves using a high-order conservative scheme modified by a strong diffusion term in order to maintain stability and produce positive values. The second stage tries to recover the lost accuracy by applying the negative of the diffusion used in the first step. The anti diffusion is applied everywhere except where it is advantageous to keep some or all of the diffusive effect of step one, such as near extrema, where oscillations might occur. The anti diffusion is applied by correcting the fluxes. The corrected fluxes are restricted so that the concentration at any point is bounded by its neigbouring points, thereby avoiding the generation of negative values and new maxima or minima, or accentuating already existing extrema. Zalesak (1979) uses a slightly different approach to that used by Boris and Book (1973). A high- order and a low-order conservative solution is computed. The high-order and low-order fluxes from these schemes are used in the flux-corrected process to determine what fraction of each solution is used at a grid point to prevent the growth of unwanted maxima and minima and to maintain positivity. The result is a family of transport algorithms capable of resolving shock fronts.
Formally the procedure described by Zalesak (1979) is as follows;
(0 Compute Fjtm, the transportive flux given by some low-order conservative monotone preserving scheme.
(ii) Compute F}?1/2, the transportive flux given by some high-order conservative scheme.
(iii) Define the anti-diffusive flux using
(iv) Compute the concentration at the next lime level using the low order fluxes with
(v) Limit the anti-diffusive flux
where 0 < otj+l/2 < 1
so that the flux corrected concentration c7n+1 is free of extrema not found
• — n + 1 n
Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.44
(vi) The corrected flux,F.+1/2 is used to update the nodal concentrations and advance the solution in time using
The essential difference between flux corrected transport schemes is in the choice of both the low- and high-order schemes and the limiter, a imposed in step (v) above. The flux corrected transport scheme can be applied using any conservative scheme. The only requirement for the low-order scheme is that it is monotone preserving. The high-order scheme is chosen to provide an adequate resolution of shocks.
Boris and Book (1973) and Zalesak (1979) have proposed procedures for limiting the flux. The flux limiter should generate no new maxima or minima in the solution, nor should it accentuate already existing extrema. Zalesak’s approach is:
To avoid accentuation of maximum and minima set
F
1 j + m
- c /* 1) < o
- c/.:1) <o
- c;;1) < o.
To avoid under and over shoots set
*7+1/2 min(R-+l,Rj) i f F J . 1/2 ^ 0 min(Rj ,/?/+1) if F , l;2 < 0 where r
:
AxminQjCj"31 - c / +1)/P/ 0 if p ; > 0 if p; = 0 andRi
Ax min(l ,Cjn*1 - c ^ / P J
0
if p; > 0
if PJ = 0
in which c"““, c j m are the upper and lower bounds on the predicted concentration,
Pj = the sum o f all antidiffusive fluxes into grid point j = max{0,F._1/2) - min(0,Fj+m)
and
Pj = the sum o f all antidiffusive fluxes away from grid point j
= max(0,F^m) - min{0,F)_1/2).
The final step required is to determine the upper bound, cj™ and the lower bound, c fm on the predicted concentration. One choice is to limit the anti-diffusive fluxes so that the concentration at any point does not exceed the value at neighboring points. For example
Cj = max{Cj.x, Cj , c;+1 ) and c, = min{Cj_x , Cj , c;>1 ).
This scheme coincides with Boris and Book (1973) limiter, which can be simply written as
^y+ i / 2 = +1/2)maxip,min(Aj {/7signAj+j/2, | ^*^21 >^/+3/2^^^^/+1/2)}
1 • 1 . n+1 n+1
in which A.+1/2 = Cj+X - c, .
Alternatively, the maximum and minimum o f six concentration values could be used. Three concentrations values from the final solution on the previous time level and three values from the newly computed low-order solution could be used to yield
c T = max(max(CjVl\cJn_l),max(cf*\cJn),max(Cj*\l,cfl)) and
c,mw = min(min(cf\l,c f x), min{cf*x,c- ) , m in { c f\\c fx)).
This scheme was proposed by Zalesak (1979) and has the advantage o f permitting the use o f any concentrations to set the maximum and minimum concentration limits.
Morrow and Noye (1992) developed a flux corrected transport algorithm for the solution o f the conservative advective-diffusion equations which is stable for all Peclet numbers, provided
Numerical Methods fo r Solving the One-Dimensional Advective-Diffusion Equation 2.46
Cr < 1. In addition, both the high- and low-order schemes possess the same stability constraints.
For the low-order scheme, Morrow and Noye (1992) used a fully implicit differencing scheme for the diffusion term, coupled with an explicit upwind differencing for the advective term to produce
-1 / Pec”.*,1 + (1 + 2IPe)c"'' - 1 = CVc", + (1 - which is stable, diagonally dominant, and has non-negative coefficients for
0 < Cr < 1 and UPe > 0.
The application of the implicit scheme necessitates the solution of a system of simultaneous equations. Fortunately the coefficient matrix is tridiagonal. If the system of equations are diagonally dominant, the very efficient Thomas algorithm can be used.
The low-order scheme has the following modified equivalent equation
dc ~dt
D + uAxjl - Cr)'
2 — + dx2 0(A x2)
which is identical to that obtained for the forward time upwind scheme, equation (2.5). It is a first-order very diffusive scheme. However, the implicit treatment of the diffusion term in this scheme means that the scheme is not limited by the stability condition l/Pe < 1/2 and is therefore more suitable for large and small diffusion than the forward time upwind scheme. This scheme is stable and diagonally dominant for
0 < Cr < 1 and UPe > 0. For the low-order scheme, the low-order flux is given by
Fi . m = ^ ( O r e ' - (c;V -
For the high-order scheme, Morrow and Noye (1992) use the fourth-order scheme of Steinle and Morrow (1989) for the advection, coupled with a second-order differencing of the diffusion term. The resulting high-order scheme for the conservative advective-diffusion scheme is
(2 - 3 Cr + Cr2 - 6/Pe)cJ_\' + (8 - 2 + 12 + (2 + 3 0 ? + Cr2 - 6IPe)c":' = (2 + 3Cr + Cr2 + 6/Pe)c/_,
+ (8 - 2 Cr2- 12/Pe)c" + (2 - 3 + +
0 < Cr < 1 and l/Pe > 0 which is identical to the low-order scheme.
The high-order scheme has the following modified equivalent partial differential equation (Morrow and Noye [1992])
Once the high- and low-order solutions and the anti-diffusive flux have been determined, either the Boris and Book (1973) or Zalesak (1979) flux limiter strategy is used.
Numerical test have shown that the Boris and Book and Zalesak limiter produce identical results for the simulation of the advection of shocks. However, for the advection of smooth profiles, the Zalesak limiter is slightly more accurate than the Boris and Book limiter. Therefore, in this study Zalesak’s flux limiter has been employed. The use of the flux corrected transport scheme of Morrow and Noye and Zalesak’s flux limiter for the simulation of the advection of the test profile is illustrated in Figure 2.16.
There is excellent resolution of the step function. Unfortunately, the flux limiter used has clipped the smooth profile. Although Zalesak describes a peak-preserving strategy, Steinle and Morrow (1989) omit the peak-preserver incorporated in Zalesak’s flux limiter scheme because it was found to produce overshoots.