高精度隐式参差光滑格式的构造

Construction of High-Order Accuracy Implicit Residual Smoothing Schemes

参照Ｌａｘ－Ｗｅｎｄｒｏｆｆ格式的构造方法，就双曲型方程、抛物型方程和双曲－抛物型方程，构造了一种新的ＩＲＳ（ｉｍｐｌｉｃｉｔｒｅｓｉｄｕａｌｓｍｏｏｔｈｉｎｇ）格式．该ＩＲＳ格式有二阶或三阶时间精度且大大地拓宽了解的稳定区域和ＣＦＬ数．这种新的ＩＲＳ格式有中心加权型和迎风偏向型两种，并用ｖｏｎ－Ｎｅｕｍａｎｎ分析方法分析了格式的稳定范围．讨论了在透平机械中广泛应用的Ｄａｗｅｓ方法的局限性，发现该方法对稳态问题得出的解与时间步长的选取有关，对粘性问题求解时，时间步长受严格限制．最后，结合 ＴＶＤ（ｔｏｔａｌ ｖａｒｉａｔｉｏｎ ｄｉｍｉｎｉｓｈｉｎｇ）格式和四阶Ｒｕｎｇｅ－Ｋｕｔｔａ技术，用 ＩＲＳ格式和 Ｄａｗｅｓ方法对二维反射激波场进行了数值模拟，数值结果支持本文的分析结论．

Referring to the construction way of Lax-Wendroff scheme, new IRS(Implicit Residual Smoothing) schemes have been developed for hyperbolic, parabolic and hyper-parabolic equations These IRS schemes IRS 2nd- or 3rd-order time accuracy, and can extend the stability region of basic explicit time-stepping scheme greatly and thus can permit higher CFL number in the calculation of flow field. The central smoothing and upwind-bias smoothing techniques have been developed too. Based on one-dimensional linear model equation, it has been found that the scheme is unconditionally stable according to the von-Neumann analysis. The limitation of Dawes' methods, which has been applied in turbomachinery widespreadly, has been discussed on solving steady flow and viscous flow. It is shown that stable solution of the method is not completely independent to the value of the step. In the end, numerical results by using IRS schemes and Dawes', method as wall as TVD (total varia- tion diminishing) scheme and four-stage Runge-Kutta technique are presented to verify the analytical conclusions.