一种求解欧拉方程的高精度数值格式

A high accuracy numerical scheme for solving Euler equation

为了优化加权本质无振荡(weighted essentially non-oscillatory,WENO)格式在流体力学中的计算性能,在WENO-JS格式和WENO-M格式加权方法研究分析的基础上,基于广义不可分差分算子推导出新局部光滑因子,构造出WENO-NS格式。将总能对流迎风和分压(E-CUSP)格式与七阶WENO-NS格式耦合得到新格式。在空间层上,采用七阶WENO-NS格式重构低耗散E-CUSP格式中的通量;在时间层上,采用四阶总变差递减的龙格-库塔(total variation decreasing Runge-Kutta,TVD-Runge-Kutta)方法推进,对一维激波管问题进行了数值模拟。观察可得,耦合的E-CUSP-WENO-NS7对接触间断和激波的捕捉更为精确。结果分析表明,耦合的新格式可以更陡峭地捕捉到激波,计算结果精度高,稳定性能好。

In order to optimize the computational performance of weighted essentially non-oscillatory(WENO)scheme in hydrodynamics,on the basis of the research and analysis of the weighted methods of WENO-JS scheme and WENO-M scheme,a new local smoothness indicator is obtained from the generalized undivided difference operator,and the WENO-NS scheme is constructed.The new scheme is obtained by coupling the total energy convective upwind and split pressure schemes(E-CUSP)with the seventh-order WENO-NS scheme.In the space layer,the flux in the low dissipation E-CUSP scheme is reconstructed by the seventh-order WENO-NS scheme,and in the time layer,the fourth-order total variation decreasing Runge-Kutta(TVD-Runge-Kutta)method is used to simulate the one-dimensional shock tube problem.It can be observed that the coupled E-CUSP-WENO-NS7 is more accurate in capturing contact discontinuities and shock waves.The results show that the new coupled scheme can capture shock waves more steeply,and the calculation results are of high accuracy and good stability.