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 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 O(N) operations, instead of standard matrix inversion, which is 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 multifluid 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.

中文翻译：

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用于刚性多流体灰尘和气体流体动力学的快速二阶求解器

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之间的动量传递（氮）的粉尘种类。该方法将流体动力学方程与隐式-显式格式相结合，并使用对角隐式、渐近稳定龙格-库塔法 (DIRK) 求解动量方程中的刚性源项。特别是，DIRK 承认一个简单的解析解，可以用以下方法进行评估： 氧（氮） 运算，而不是标准矩阵求逆，即 氧（氮）3 。因此，解析解显着降低了多流体方法的计算成本，使其适合研究具有粒度分布的系统的动力学。我们证明该方法可以保持机器精度动量，并在外部加速度恒定的情况下收敛到正确的平衡解。为了验证我们的数值方法，我们提出了一系列简单的流体动力学测试，包括声波阻尼、粉尘冲击、多流体粉尘牛仔裤不稳定性以及稳态气体-粉尘漂移计算。MDIRK 的简单性为构建快速高阶、渐近稳定的多流体方法奠定了基础。