Bolis A, Cantwell C D, Kirby R M, Sherwin S J
Department of Aeronautics, Imperial College London South Kensington Campus, London, UK.
School of Computing, University of Utah Salt Lake City, UT, USA.
Int J Numer Methods Fluids. 2014 Jul 20;75(8):591-607. doi: 10.1002/fld.3909. Epub 2014 Apr 11.
We investigate the relative performance of a second-order Adams-Bashforth scheme and second-order and fourth-order Runge-Kutta schemes when time stepping a 2D linear advection problem discretised using a spectral/ element technique for a range of different mesh sizes and polynomial orders. Numerical experiments explore the effects of short (two wavelengths) and long (32 wavelengths) time integration for sets of uniform and non-uniform meshes. The choice of time-integration scheme and discretisation together fixes a CFL limit that imposes a restriction on the maximum time step, which can be taken to ensure numerical stability. The number of steps, together with the order of the scheme, affects not only the runtime but also the accuracy of the solution. Through numerical experiments, we systematically highlight the relative effects of spatial resolution and choice of time integration on performance and provide general guidelines on how best to achieve the minimal execution time in order to obtain a prescribed solution accuracy. The significant role played by higher polynomial orders in reducing CPU time while preserving accuracy becomes more evident, especially for uniform meshes, compared with what has been typically considered when studying this type of problem.© 2014. The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons, Ltd.
我们研究了二阶亚当斯-巴什福斯格式以及二阶和四阶龙格-库塔格式在对二维线性平流问题进行时间步长推进时的相对性能，该二维线性平流问题采用谱/单元技术进行离散，涉及一系列不同的网格尺寸和多项式阶数。数值实验探讨了针对均匀和非均匀网格集进行短时间（两个波长）和长时间（32个波长）时间积分的影响。时间积分格式和离散化的选择共同确定了一个CFL限制，该限制对最大时间步长施加了限制，可用于确保数值稳定性。步数以及格式的阶数不仅会影响运行时间，还会影响解的精度。通过数值实验，我们系统地突出了空间分辨率和时间积分选择对性能的相对影响，并提供了关于如何最佳地实现最短执行时间以获得规定解精度的一般指导原则。与研究此类问题时通常考虑的情况相比，高阶多项式在保持精度的同时减少CPU时间方面所起的重要作用变得更加明显，特别是对于均匀网格而言。© 2014. 作者。《国际流体数值方法杂志》由约翰·威利父子有限公司出版。
