剪切盒近似在Athena中的实现
Implementation of the Shearing Box Approximation in Athena
摘要：
本文描述了在Athena磁流体动力学（MHD）程序中研究吸积盘动力学的剪切盒近似的实现。二阶Crank-Nicholson时间差分用于动量方程中出现的科里奥利和潮汐重力源项，以确保精度和稳定性。我们表明，这种方法保存能量的周转振荡的水动力流舍入误差。在能量方程中，潮汐重力源项被差分为有效势的梯度，以保证总能量（包括重力势能）也守恒到舍入误差。我们介绍了一种基于约束输运的轨道平流算法，以保持对磁场的无发散约束。该算法去除了时间步长对轨道速度的约束，使截断误差在径向位置上更加均匀。为了保持总的垂直磁通量，有必要对径向边界处应用的剪切盒边界条件进行修改。在原则上，类似的修正也需要保存质量，动量和能量，但在实践中，我们发现轨道平流方法保存这些数量超过0.03%超过数百个轨道。该算法已被应用于研究的非线性制度的MRI在非常广泛的（高达32标度高度）的水平域。
英文摘要：
We describe the implementation of the shearing box approximation for the study of the dynamics of accretion disks in the Athena magnetohydrodynamics (MHD) code. Second-order Crank-Nicholson time differencing is used for the Coriolis and tidal gravity source terms that appear in the momentum equation for accuracy and stability. We show this approach conserves energy for epicyclic oscillations in hydrodynamic flows to round-off error. In the energy equation, the tidal gravity source terms are differenced as the gradient of an effective potential in a way which guarantees that total energy (including the gravitational potential energy) is also conserved to round-off error. We introduce an orbital advection algorithm for MHD based on constrained transport to preserve the divergence-free constraint on the magnetic field. This algorithm removes the orbital velocity from the time step constraint, and makes the truncation error more uniform in radial position. Modifications to the shearing box boundary conditions applied at the radial boundaries are necessary to conserve the total vertical magnetic flux. In principle similar corrections are also required to conserve mass, momentum and energy, however in practice we find the orbital advection method conserves these quantities to better than 0.03% over hundreds of orbits. The algorithms have been applied to studies of the nonlinear regime of the MRI in very wide (up to 32 scale heights) horizontal domains.
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