用于刚性多流体尘埃与气体流体动力学的快速二阶求解器
A Fast second-order solver for stiff multifluid dust and gas hydrodynamics
AI总结：
本文提出MDIRK多流体二阶对角隐式龙格-库塔方法，利用O(N)解析解替代O(N)^3矩阵求逆求解刚性动量源项，显著降低计算成本，并通过多种流体动力学测试验证其守恒性与收敛性。
摘要：
我们提出MDIRK：一种多流体二阶对角隐式龙格-库塔方法，用于研究气体与任意数量（N）尘埃种类之间的动量传递。该方法采用隐式-显式（IMEX）格式对流体动力学方程进行积分，并利用对角隐式渐近稳定的龙格-库塔方法（DIRK）求解动量方程中的刚性源项。特别地，DIRK允许一个简单的解析解，其计算复杂度为O(N)次运算，而非标准矩阵求逆的O(N)^3。因此，该解析解显著降低了多流体方法的计算成本，使其适用于研究具有粒径分布的系统动力学。我们证明该方法能将动量守恒到机器精度，并在恒定外部加速度下收敛到正确的平衡解。为验证我们的数值方法，我们给出了一系列简单的流体动力学测试，包括声波阻尼、尘埃激波、多流体尘埃金斯不稳定性以及稳态气体-尘埃漂移计算。MDIRK的简洁性为构建快速高阶渐近稳定的多流体方法奠定了基础。
英文摘要：
We present MDIRK: a Multifluid second-order Diagonally-Implicit Runge-Kutta method to study momentum transfer between gas and an arbitrary number ($N$) of dust species. The method integrates the equations of hydrodynamics with an Implicit Explicit (IMEX) scheme and solves the stiff source term in the momentum equation with a diagonally-implicit asymptotically stable Runge-Kutta method (DIRK). In particular, DIRK admits a simple analytical solution that can be evaluated with $\mathcal{O}(N)$ operations, instead of standard matrix inversion, which is $\mathcal{O}(N)^3$. Therefore the analytical solution significantly reduces the computational cost of the multifluid method, making it suitable for studying the dynamics of systems with particle-size distributions. We demonstrate that the method conserves momentum to machine precision and converges to the correct equilibrium solution with constant external acceleration. To validate our numerical method we present a series of simple hydrodynamic tests, including damping of sound waves, dusty shocks, a multi-fluid dusty Jeans instability, and a steady-state gas-dust drift calculation. The simplicity of MDIRK lays the groundwork to build fast high-order asymptotically stable multifluid methods.
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