一种基于AUSM分裂的真正多维HLL格式

文章给出了一种真正多维的HLL Riemann解算器.采用AUSM分裂将通量分解成为对流通量和压力通量, 其中对流通量的计算采用迎风格式, 压力通量的计算采用HLL格式, 且将HLL格式的耗散项中的密度差用压力差代替, 从而使得格式能够分辨接触间断.为了实现数值格式真正多维的特性, 分别计算了网格界面中点和角点上的数值通量, 并且采用Simpson公式加权组合中点和角点上的数值通量得到网格界面的数值通量.为了减少重构角点处状态时的模板宽度, 计算中采用基于SDWLS梯度的线性重构获得2阶空间精度, 而时间离散采用2阶保强稳Runge-Kutta方法.数值实验表明, 相比于传统的一维HLL格式, 文章的真正多维HLL格式具有能够分辨接触间断, 以及更大的时间步长等优点.与其他能够分辨接触间断的格式(例如HLLC格式)不同, 真正多维的HLL格式在计算二维问题时不会出现激波不稳定现象. 

A Genuinely Multidimensional HLL Riemann Solver Based on AUSM Splitting

A genuinely multidimensional HLL Riemann solver was given. The flux vector of the Euler equations was split into convection and pressure parts. The convection part was evaluated by using the upwind scheme and the pressure part was evaluated by using a modified HLL scheme. In the modified HLL scheme, the pressure difference was replaced by density difference in the dissipative term in order to capture the contact accurately. In order to obtain the genuinely multidimensional property, the numerical fluxes at the midpoint and the two corners of the cell interface were evaluated respectively, and the Simpson rule was used to obtain the final numerical flux through the interface. The linear reconstruction using SDWLS gradients was used for second order spatial accuracy, and the time derivative was discretized using the second order strong stability preserving Runge-Kutta method. Compared with the traditional one dimensional HLL scheme, the genuinely multidimensional HLL scheme can capture the contact discontinuity, and can use larger time step. Unlike other schemes which can capture the contact discontinuity accurately such as the HLLC scheme, the genuinely multidimensional HLL scheme eliminates the phenomena of numerical shock instability in 2D cases.