基于HLLC格式的理想MHD方程数值模拟

针对理想MHD(magnetohydrodynamics)方程数值求解困难的问题，基于原始的HLLC (Harten-Lax-Van Leer Contact wave)近似黎曼解方法，发展出一种新的适用于理想MHD问题的通量计算格式。控制方程采用有限体积法离散，时间推进采用隐式的LU-SGS（lower-upper symmetric Gauss-Seidel）格式，并且采用双曲型散度清除技术来抑制磁场散度的累积。通过一维激波管问题的数值模拟表明，HLLC-MHD格式能准确的分辨并捕捉复杂的磁流体力学波系结构，保证数值计算的精度，最大的数值计算误差不超过10%；通过二维的Rotor问题的数值模拟研究表明，HLLC-MHD格式能够应用于多维理想MHD问题的数值模拟，并且能够准确捕捉多维情况下磁流场中的阿尔文波；通过对比有、无散度清除的Rotor问题计算结果表明，双曲型散度清除技术可以将磁场散度峰值从50降低到2，有效抑制高磁场散度区域的散度累积，同时也会将误差传播到低散度区域，并且引起边界处散度的累积，影响计算的稳定性。 

Numerical simulation of ideal MHD equations using HLLC scheme

A new flux calculating method based on the original HLLC (Harten-Lax-Vanleer contact wave) approximate riemann solver for ideal magnetohydrodynamics (MHD) equations was developed. The governing equations were discretized by finite volume method, and the time integration employed the implicit LU-SGS (lower-upper symmetric Gauss-Seidel) scheme. A hyperbolic divergence cleaning technique was introduced to suppress divergence of magnetic field. The numerical results of 1-D MHD shock-tube problem showed that HLLC-MHD scheme could accurately capture the magnetohydrodynamics waves, and the maximum numerical error was not more than 10%. The results of 2-D Rotor problem showed that HLLC-MHD scheme could be used to simulate the multi-dimension ideal MHD problem and accurately capture the Alfven wave of magnetic field. By comparing the results with and without divergence cleaning technique, the hyperbolic divergence cleaning technique could drop the peak value of magnetic field divergence from 50 to 2, and restrain the magnetic field divergence accumulation in the high divergence region, but also propagate the error to the low divergence region and cause the accumulation of the divergence at the boundary. It will influence the stability of the calculation.