Upwinding, QUICK and SMART

To fully appreciate the advantages of the SMART algorithm it is necessary to go into a few details. Initially a few definitions will be introduced and then the problems of the algorithms of upwinding and QUICK will be explained. Finally the algorithm SMART will be described and a brief description given of how it works. For further details on the formulation of these algorithms consult the references given in Selected References.

The problem is to estimate a sub-nodal value of the concentration such that:

• the concentration values are within the range of possible values

• the estimate is as accurate as possible

• the algorithm is computationally efficient

Definitions

• Bounded. This effectively means that the value calculated for the sub-nodal concentration is realistic (i.e. it is within the two surrounding nodal values and this implies that it is always within the range of realistic concentrations).
• Accuracy. Classically, the accuracy of finite difference approximations is judged in terms of the leading truncation error term of a Taylor series expansion. However an increase in the formal order of accuracy does not necessarily improve the approximation.
• Normalization. Given , and are concentrations upstream, central and downstream, then the normalised coordinates are given by:

                                       (3)

Hence =1 and . We are interested in the concentration mid way between the central and downstream nodes whose normalised value is denoted by .

• Numerical Diffusion. A tendency for sharp concentration gradients to become less steep due to the numerical approximations used.

Problems with Upwinding and QUICK

The upwinding algorithm is a first order algorithm.

                                       (4)

This algorithm is unconditionally bounded however it suffers from numerical diffusion and becomes very inaccurate with time.

The QUICK algorithm (Quadratic Upstream Interpolation for Convective Kinematics) is designed to overcome the inaccuracies of first order upwinding. It is third order accurate. However it is not bounded and tends to "overshoot" and perform damped oscillations about the required concentration value whenever there is a sharp increase or decrease in concentration levels.This means that the method produces unrealistic concentration values (e.g. negative concentrations).

                       (5)

SMART

In their paper on the analysis of these systems, Gaskell and Lau (1988) reason that provided the value of is between that of and then the Quick algorithm is bounded and, by a monotonic argument, is a desirable choice of algorithm for this range of . They also reasoned that if is outside this region then the only unconditionally bounded algorithm is first order upwinding.

Therefore they produced a new type of algorithm called SMART (Sharp and Monotonic Algorithm for Realistic Transport by convection), which combines the best qualities of the two algorithms, Upwinding and QUICK, by patching them together with regard to continuity

In normalised form, the algorithm is:

                                                                     (6)

This immediately becomes more complex in unnormalised variables, (with all values calculated at one time):

                                       (7)

There are a number of benefits for using SMART, these are:

• The SMART algorithm preserves boundedness and precludes the occurrence of spurious spatial oscillations while maintaining a high degree of accuracy.It is based on sound physical arguments and strict adherence to a convection boundedness criterion.

• The SMART algorithm results in a relatively low level of numerical diffusion and preserves steep gradients without accentuating or creating any new maxima or minima.

• Although SMART is non-linear the computational cost associated with the use of this scheme is found to be relatively low.

• The simulation of water quality using SMART, as compared with Upwinding or QUICK, is considerably more realistic.