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首页> 外文期刊>Computers & Fluids >Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows
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Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows

机译：线性隐式Rosenbrock型Runge-Kutta方案应用于可压缩和不可压缩非恒定流的间断Galerkin解

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摘要

In this work we investigate the use of linearly implicit Rosenbrock-type Runge-Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the Navier-Stokes equations. The final goal of this activity is the application of such schemes to the high-order accurate, both in space and time, simulation of turbulent flows. Besides being able to overcome the severe time step restriction of explicit schemes, Rosenbrock schemes have the attractive feature of requiring just one Jacobian matrix evaluation per time step, thus reducing the overall computational effort. Several high-order (up to sixth order) Rosenbrock schemes available in the literature have been implemented and evaluated on benchmark test cases of both compressible and incompressible flows. For the sake of completeness, the sets of coefficients of the schemes here considered have been reported in an Appendix to the paper. An implementation of Rosenbrock schemes for systems of equations with a solution dependent block diagonal matrix multiplying the time derivative is here proposed and described in detail. This can occur, for example, if sets of working variables different from the conservative ones are used in the compressible Navier-Stokes equations. In particular, we have found useful to employ primitive variables based on the logarithms of pressure and temperature in order to ensure the positivity of all thermodynamic variables at the discrete level. The best performing Rosenbrock scheme resulting from our analysis has then been applied to the Implicit Large Eddy Simulation of the transitional flow around the Selig-Donovan SD7003 airfoil. (C) 2015 Elsevier Ltd. All rights reserved.
机译：在这项工作中，我们研究了使用线性隐式Rosenbrock型Runge-Kutta方案对Navier-Stokes方程的时间高阶非连续Galerkin空间离散进行积分。这项活动的最终目标是将此类方案应用到在空间和时间上模拟湍流的高阶精确度。 Rosenbrock方案除了能够克服显式方案对时间步长的严格限制外，还具有吸引人的特征，即每个时间步长仅需进行一次雅可比矩阵评估，从而减少了总体计算量。在可压缩和不可压缩流的基准测试用例上，已经实现并评估了文献中可用的几种高阶（最高六阶）Rosenbrock方案。为了完整起见，本文附录中已报告了此处考虑的方案的系数集。本文提出并详细描述了方程组的Rosenbrock方案的实现，该方程组具有与解相关的块对角矩阵乘以时间导数。例如，如果在可压缩的Navier-Stokes方程中使用了不同于保守变量的工作变量集，则会发生这种情况。特别地，我们发现基于压力和温度的对数采用原始变量很有用，以确保离散水平上所有热力学变量的正性。然后，将我们分析得出的性能最好的Rosenbrock方案应用于Selig-Donovan SD7003机翼周围过渡流的隐式大涡模拟。 （C）2015 Elsevier Ltd.保留所有权利。

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