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Optimal time advancing dispersion relation preserving schemes

机译：最佳时间提前色散关系保持方案

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摘要

In this paper we examine the constrained optimization of explicit Runge-Kutta (RK) schemes coupled with central spatial discretization schemes to solve the one-dimensional convection equation. The constraints are defined with respect to the correct error propagation equation which goes beyond the traditional von Neumann analysis developed in Sengupta et al. [T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: beyond von Neumann analysis, J. Comput. Phys. 226 (2007) 1211-1218]. The efficiency of these optimal schemes is demonstrated for the one-dimensional convection problem and also by solving the Navier-Stokes equations for a two-dimensional lid-driven cavity (LDC) problem. For the LDC problem, results for Re=1000 are compared with results using spectral methods in Botella and Peyret [O. Botella, R. Peyret, Benchmark spectral results on the lid-driven cavity flow, Comput. Fluids 27 (1998) 421-433] to calibrate the method in solving the steady state problem. We also report the results of the same flow at Re=10,000 and compare them with some recent results to establish the correctness and accuracy of the scheme for solving unsteady flow problems. Finally, we also compare our results for a wave-packet propagation problem with another method developed for computational aeroacoustics.
机译：在本文中，我们研究了显式Runge-Kutta（RK）方案与中央空间离散化方案的约束优化，以解决一维对流方程。约束是针对正确的误差传播方程定义的，该方程超出了Sengupta等人开发的传统von Neumann分析。 [T.K. Sengupta，A。Dipankar，P。Sagaut，错误动态：超越冯·诺依曼分析，J。Comput。物理226（2007）1211-1218]。通过一维对流问题以及求解二维盖驱动腔（LDC）问题的Navier-Stokes方程，证明了这些最优方案的效率。对于最不发达国家问题，将Re = 1000的结果与Botella和Peyret中使用光谱方法的结果进行比较[O. Botella，R。Peyret，盖驱动腔流动的基准光谱结果，计算。流体27（1998）421-433]以校正解决稳态问题的方法。我们还报告了Re = 10,000时相同流量的结果，并将它们与最近的一些结果进行比较，以建立解决不稳定流量问题的方案的正确性和准确性。最后，我们还将波包传播问题的结果与为计算航空声学开发的另一种方法进行比较。

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