一种基于TV分裂的真正多维Riemann解法器

A Genuinely Multidimensional Riemann Solver Based on the TV Splitting

给出了一种真正多维的HLL Riemann解法器.采用TV(Toro-Vázquez)分裂将通量分裂成对流通量和压力通量,其中对流通量的计算采用类似于AUSM格式的迎风方法,压力通量的计算采用波速基于压力系统特征值的HLL格式,并将HLL格式耗散项中的密度差用压力差代替,来克服传统的HLL格式不能分辨接触间断的缺点.为了实现数值格式真正多维的特性,分别计算网格界面中点和角点上的数值通量,并且采用Simpson公式加权中点和角点上的数值通量来得到网格界面上的数值通量.采用基于SDWLS(solution dependent weighted least squares)梯度的线性重构来获得空间的二阶精度,时间离散采用二阶Runge-Kutta格式.数值实验表明,相比于传统的一维HLL格式,该文的真正多维HLL格式具有能够分辨接触间断,消除慢行激波波后振荡以及更大的时间步长等优点.并且,与其他能够分辨接触间断的格式(例如HLLC格式)不同的是,真正多维的HLL格式在计算二维问题时不会出现数值激波不稳定现象.

A genuinely multidimensional HLL Riemann solver was given.The flux vector of the Euler equations was split into convection and pressure parts based on the TV splitting method.The convection part was evaluated by means of the upwind method similar to the AUSM scheme,and the pressure part was evaluated with a modified HLL scheme.In the modified HLL scheme,the choices of wave speed were based on the pressure system rather than the Euler equations,and the pressure difference was replaced by the density difference in the dissipative term in order to capture the contact accurately.To obtain the genuinely multidimensional property,the numerical fluxes at the midpoint and the 2 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 based on the SDWLS gradients was implemented for 2nd-order spatial accuracy,and the time derivative was discretized with the 2nd-order Runge-Kutta method.Compared with the traditional 1D HLL scheme,the genuinely multidimensional HLL scheme can effectively capture the contact discontinuity,and use larger time steps.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.