Computer evaluation of high order numerical schemes to solve advective transport problems

Abstract

This study evaluates the performance of three representative high-order finite difference schemes to solve two sets of simple one-dimensional benchmark problems in terms of their ability to resolve spurious oscillation, numerical spreading, and peak clipping. Three models, namely QUICKEST, ULTIMATE, and ENO were constructed to represent the classical high-order schemes without a flux limiter, TVD with a flux limiter, and TVB schemes, respectively. Three sets of results generated by QUICKEST, ULTIMATE, and ENO were compared with the analytical solutions. The first set indicated that none of these high-order schemes could yield satisfactory simulations when the grid size and time-step size specified 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 deteriorated even with the reduced grid size and time-step size when 100 folds of simulation times was conducted. The ENO and ULTIMATE schemes successfully eliminated spurious oscillations for all cases as expected. The QUICKEST scheme alleviated the problem of spurious oscillations only when the reduced grid and time-step sizes were used. In terms of numerical spreading and peak clipping, none of the three schemes produced satisfactory results unless the reduced grid and time-step were used. Peak clipping poses a more severe problem for these high order schemes than numerical spreading. A common set of benchmark problems is needed for the evaluation and testing of any numerical scheme.