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Energy-conserving runge-kutta methods for the incompressible navier-stokes equations

机译：不可压缩纳维斯托克斯方程的节能Runge-kutta方法

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摘要

Energy-conserving methods have recently gained popularity for the spatial discretization of the incompressible Navier-Stokes equations. In this paper implicit Runge-Kutta methods are investigated which keep this property when integrating in time. Firstly, a number of energy-conserving Runge-Kutta methods based on Gauss, Radau and Lobatto quadrature are constructed. These methods are suitable for convection-dominated problems (such as turbulent flows), because they do not introduce artificial diffusion and are stable for any time step. Secondly, to obtain robust time-integration methods that work also for stiff problems, the energy-conserving methods are extended to a new class of additive Runge-Kutta methods, which combine energy conservation with L-stability. In this class, the Radau IIA/B method has the best properties. Results for a number of test cases on two-stage methods indicate that for pure convection problems the additive Radau IIA/B method is competitive with the Gauss methods. However, for stiff problems, such as convectiondominated flows with thin boundary layers, both the higher order Gauss and Radau IIA/B method suffer from order reduction. Overall, the Gauss methods are the preferred method for energy-conserving time integration of the incompressible Navier-Stokes equations.
机译：节约能量的方法最近在不可压缩的Navier-Stokes方程的空间离散化中获得了普及。在本文中，研究了隐式的Runge-Kutta方法，该方法在及时积分时保留了此属性。首先，构造了多种基于高斯，拉多和洛巴托正交的节能朗格库塔方法。这些方法适用于以对流为主的问题（例如湍流），因为它们不会引入人为扩散并且在任何时间步均稳定。其次，为了获得适用于严峻问题的鲁棒的时间积分方法，将节能方法扩展到一类新的加性Runge-Kutta方法，该方法将节能与L稳定性相结合。在此类中，Radau IIA / B方法具有最佳的性能。关于两阶段方法的大量测试案例的结果表明，对于纯对流问题，加法Radau IIA / B方法与高斯方法具有竞争力。但是，对于刚性问题，例如具有薄边界层的以对流为主的流动，高阶高斯方法和Radau IIA / B方法都存在阶数减少的问题。总体而言，高斯方法是不可压缩Navier-Stokes方程的节能时间积分的首选方法。

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