COMPUTER EVALUATION OF HIGH-ORDER NUMERICAL SCHEMES TO SOLVE ADVECTIVE TRANSPORT PROBLEMS

This study evaluates the performance of three representative high-orde r finite difference schemes to solve two sets of simple one-dimensiona l benchmark problems in terms of their ability to resolve spurious osc illation, numerical spreading, and peak clipping. Three models, namely QUICKEST, ULTIMATE, and ENO were constructed to represent the classic al high-order schemes without a flux limiter, TVD with a flux limiter, and TVB schemes, respectively. Three sets of results generated by QUI CKEST, ULTIMATE, and ENO were compared with the analytical solutions. The first set indicated that none of these high-order schemes could yi eld satisfactory simulations when the grid size and time-step size spe cified by the benchmark problems were used. The second set showed that all three numerical schemes generated excellent computations when the grid size was reduced to one-tenth and the time-step size was reduced to one-fifth of those specified by the benchmark problems. The third set demonstrated that the results obtained by these schemes deteriorat ed even with the reduced grid size and time-step size when 100 folds o f simulation times was conducted. The ENO and ULTIMATE schemes success fully eliminated spurious oscillations for all cases as expected. The QUICKEST scheme alleviated the problem of spurious oscillations only w hen the reduced grid and time-step sizes were used. In terms of numeri cal spreading and peak clipping, none of the three schemes produced sa tisfactory results unless the reduced grid and time-step were used. Pe ak clipping poses a more severe problem for these high order schemes t han numerical spreading. A common set of benchmark problems is needed for the evaluation and testing of any numerical scheme.