Preliminary Study And Improve Of The ENO/WENO Schemes In Riemann Problems

Numerical simulation has becoming a coordinate solution of theoretic and laboratorial method. However, current fluid simulations are increasingly complex. Traditional meth-ods will cost lots of computer’s computing time, which results in computing inefficiency. Hence, it’s necessary to optimize the numerical schemes presuppose that no extra numer-ical oscillation in the scheme. In this paper, we optimized one-dimensional ENO/WENO schemes and two-dimensional ENO schemes for Riemann problems so that achieving the purpose of saving computing time without extra numerical oscillation. The main contents and conclusions are summarized as follows:Firstly, traditional ENO scheme need to calculate three stencil polymerization when solving the intermittent problems of shock capturing, which are quite different only in the vicinity of discontinuities while are smooth in other positions. For this feature, we optimized ENO schemes. First, we determine the discontinuous and the smooth position. Next, we use the standard ENO scheme in discontinuous position and compute one stencil polymerization by optimized ENO scheme in the smooth position. Thus we can reach the purposes of both saving the computing time and nearly having no effect on the computing results. Based on this idea, we take one-dimensional shock tube as an example. Then pressure gradient is taken as a criterion to judge the smooth and discontinuous positions. Consequently, we compute the results with the standard ENO scheme where the pressure gradient is greater than the default critical value and with the preinstall polymerization at the smooth position. The results show that the computing time of the optimized schemes is shorter than the original schemes by 30%-50%. In addition, the pressure gradient is greater than 10-2 only at a very few points, Thus we set it as the default critical value to determine the discontinuous positions. The preinstall polymerization at the leftmost or rightmost stencil is selected as the final result directly according to the upwind character.Secondly, we make further optimization on WENO schemes based on optimized ENO scheme. With adjacent points’pressure gradient difference instead of direct judgment on the pressure gradient. The results show that the pressure gradient magnitude of the differencc as a criterion of optimization WENO scheme can save computing time, the optimized computing time saves 34%-52% than the traditional one. In addition, with the random constant ε as the critical value of the pressure gradient difference for judgment, both can ensure the end result generate no additional numerical oscillation,and achieve optimization purposes.Finally, we do optimized preliminary inquiry by solving a two-dimensional Riemann problem ENO scheme on 100×100 grid points.When intermittent position to judge, we must consider the two x, y directions’pressure gradient difference.Through numerical examples of two-dimensional Riemann problem, it shows nearly half(48%) of computing times can be solved, witch make the basis on the next step of shock-tube capturing more accurate.