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首页> 外文期刊>Journal of Computational Physics >Nonuniform time-step Runge-Kutta discontinuous Galerkin method for Computational Aeroacoustics
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Nonuniform time-step Runge-Kutta discontinuous Galerkin method for Computational Aeroacoustics

机译：计算航空声学的非均匀时间步长Runge-Kutta不连续Galerkin方法

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摘要

With many superior features, Runge-Kutta discontinuous Galerkin method (RKDG), which adopts Discontinuous Galerkin method (DG) for space discretization and Runge-Kutta method (RK) for time integration, has been an attractive alternative to the finite difference based high-order Computational Aeroacoustics (CAA) approaches. However, when it comes to complex physical problems, especially the ones involving irregular geometries, the time step size of an explicit RK scheme is limited by the smallest grid size in the computational domain, demanding a high computational cost for obtaining time accurate numerical solutions in CAA. For computational efficiency, high-order RK method with nonuniform time step sizes on nonuniform meshes is developed in this paper. In order to ensure correct communication of solutions on the interfaces of grids with different time step sizes, the values at intermediate-stages of the Runge-Kutta time integration on the elements neighboring such interfaces are coupled with minimal dissipation and dispersion errors. Based upon the general form of an explicit p-stage RK scheme, a linear coupling procedure is proposed, with details on the coefficient matrices and execution steps at common time-levels and intermediate time-levels. Applications of the coupling procedures to Runge-Kutta schemes frequently used in simulation of fluid flow and acoustics are given, including the third-order TVD scheme, and low-storage low dissipation and low dispersion (LDDRK) schemes. In addition, an analysis on the stability of coupling procedures on a nonuniform grid is carried out. For validation, numerical experiments on one-dimensional and two-dimensional problems are presented to illustrate the stability and accuracy of proposed nonuniform time-step RKDG scheme, as well as the computational benefits it brings. Application to a one-dimensional nonlinear problem is also investigated.
机译：Runge-Kutta不连续Galerkin方法（RKDG）具有许多优越的功能，它采用不连续Galerkin方法（DG）进行空间离散化，并使用Runge-Kutta方法（RK）进行时间积分，它已成为基于有限差分的高次幂法的有吸引力的替代方法。订购计算航空声学（CAA）方法。但是，当涉及到复杂的物理问题时，尤其是涉及不规则几何的物理问题，显式RK方案的时间步长受计算域中最小网格尺寸的限制，在获得时间精确数值解时需要较高的计算成本CAA。为了提高计算效率，本文开发了在非均匀网格上具有不均匀时间步长的高阶RK方法。为了确保解决方案在具有不同时间步长的网格的接口上正确通信，将Runge-Kutta时间积分的中间阶段在邻近此类接口的元素上的值与最小的耗散和色散误差耦合在一起。基于显式p级RK方案的一般形式，提出了一种线性耦合程序，并详细介绍了常见时间级别和中间时间级别的系数矩阵和执行步骤。给出了将耦合过程应用于经常在流体流动和声学模拟中使用的Runge-Kutta方案，包括三阶TVD方案以及低存储，低耗散和低色散（LDDRK）方案。另外，对非均匀网格上耦合过程的稳定性进行了分析。为了进行验证，提出了针对一维和二维问题的数值实验，以说明所提出的非均匀时间步长RKDG方案的稳定性和准确性，以及所带来的计算优势。还研究了在一维非线性问题中的应用。

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