针对磁流体动力学方程,本文提出了一种基于MUSCL-Hancock方法求解MHD方程的熵相容格式(EC-MHM格式),获得了一种求解理想磁流体动力学方程的高分辨率熵相容格式。该格式在解的光滑区域具有高精度;在解的间断区域可以合理地控制耗散,使抹平现象得到改善,还可有效避免非物理现象的产生。文中还证明了熵相容格式的收敛性。采用熵稳定格式、熵相容格式和新的高分辨率熵相容格式对理想磁流体动力学方程进行数值模拟。结果表明:新格式能准确地捕捉解的结构,且具有无振荡、高分辨、鲁棒等特性。Focusing on the idea magnetohydrodynamic (MHD) equations, this paper presents an entropy-consistent scheme based on the MUSCL-Hancock method for solving MHD equations, termed the Entropy Consistent MUSCL-Hancock (EC-MHM) scheme, and thus achieving a high-resolution entropy-consistent formulation for solving ideal MHD equations. This scheme exhibits high accuracy in smooth regions of the solution and effectively controls dissipation in discontinuous zones, leading to an improvement in the smearing phenomenon and efficiently preventing the emergence of non-physical oscillations. The convergence of the entropy consistent scheme is also proved. The ideal MHD equations are numerically simulated by entropy stable scheme, entropy consistent scheme and the new high resolution entropy consistent scheme. The results show that the new scheme can accurately capture the structure of the solution, and has the characteristics of no oscillation, high resolution and robustness.
Ham ilton-Jacob i方程经常应用于控制论和微分对策等方面.它与双曲型守恒律有非常紧密的联系,在一维的情形下,两者几乎完全相同.在多维的空间中,类似的情形依然存在.因此,可以通过这种联系来设计不同的近似算法来求解Ham ilton-Jacob i方程.本文利用CWENO算法成功地求解了包括控制最优化问题在内的许多标量问题,结果表明,这种算法在解的光滑区域具有很高的精度,并能很好地解决具有不连续偏导数的计算问题,数值算例结果也表明这种算法是收敛的.
