Constraint in Shock-Capturing Magnetohydrodynamics Codes
Constraint in Shock-Capturing Magnetohydrodynamics Codes
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发表时间：
2007
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通讯作者：
G. Tóth
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作者：
G. Tóth
比较了7种在数值上保持rB=0约束的方案。所有这些算法都可以与激波捕捉G-odunov基格式相结合。它们分为三类：八波格式保持对截断误差的约束，投影格式通过投影磁场在某种离散化中加强约束，而五种不同版本的约束传输/中心差分型格式对每个网格单元在某种离散化中保持r B的机器精度。结果表明，最近提出的三种约束输运算法都可以转化为纯有限体积格式，而不必交错r表示磁场。介绍了另外两种新的简单的基于中心差分的算法。较详细地讨论了投影格式的性质，并证明了即使对于不连续解，它也具有与基本格式相同的精度阶。用共轭梯度型迭代法描述了投影格式的一种灵活而有效的实现方法。给出了所有方案对阻性MHD、轴对称性和非笛卡尔GR ID的推广。在理论讨论之后，进行了数值试验，可以直接比较这七种格式和基本SCH EME的稳健性、精度和效率。所有的模拟都是用通用的Advec Code完成的，其中实现了几种冲击波捕获基本方案。虽然八波公式通常是正确的，但其中一个数值试验表明，它的非保守性偶尔会引起Inco rre t跳跃越过强烈的不连续。在大量试验的基础上，新的中心差分格式之一的投影CTI格式和构造的传输方案之一被发现是检验方法中最准确和可靠的。
Seven schemes to maintain the r B = 0 constraint numerically are compared. All these algorithms can be combined with shock-capturing G odunov type base schemes. They fall into three categories: the eight-wave formulationmaintains the constraint to truncation error, the projection schemenforces the constraint in some discretization by projecting the magnetic field, while the fi ve different versions of the constrained transport/central difference type schemes conserve r B to machine accuracy in some discretization for every grid cell . It is shown that the three constrained transport algorithms, which have been in troduced recently, can be recast into pure finite volume schemes, and the staggered r presentation of the magnetic field is unnecessary. Another two new and simple cen tral difference based algorithms are introduced. The properties of the projectio n scheme are discussed in some detail, and I prove that it has the same order of accuracy as the base scheme even for discontinuous solutions. I describe a flexible and effici ent implementation of the projection scheme using conjugate gradient type iterative methods. Generalizations to resistive MHD, to axial symmetry, and to non-Cartesian gr ids are given for all schemes. The theoretical discussion is followed by numerical tests, where the robustness, accuracy, and efficiency of the seven schemes and the base sch eme can be directly compared. All simulations are done with the Versatile Advec tion Code, in which several shock-capturing base schemes are implemented. Alt hough the eight-wave formulation usually works correctly, one of the numerical t ests demonstrates that its non-conservative nature can occasionally produce inco rre t jumps across strong discontinuities. Based on a large number of tests, the proje cti n scheme, one of the new central difference based schemes, and one of the constra ined transport schemes are found to be the most accurate and reliable among the exami ned methods.