Einkemmer Lukas, Ostermann Alexander
Department of Mathematics, University of Innsbruck, Austria.
Comput Math Appl. 2015 Feb;69(3):170-179. doi: 10.1016/j.camwa.2014.12.004.
In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley-Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme.
在本文中，我们研究了在需要高精度的情况下，用于平流方程的数值格式的误差传播。所考虑的数值方法基于快速傅里叶变换、多项式插值（使用拉格朗日或样条插值的半拉格朗日方法）以及间断伽辽金半拉格朗日方法（该方法是守恒的，并且每个单元必须存储多个值）。通过进行数值实验，我们证明了文献中给出的最坏情况误差估计能够很好地解释基于插值的半拉格朗日方法的误差传播。然而，对于间断伽辽金半拉格朗日方法，我们发现未观察到半拉格朗日误差估计的特征性质（即误差与时间步数成比例增加这一事实）。我们对这种行为给出了解释，并进行了数值模拟，证实了这两种半拉格朗日方法各自误差的不同定性特征。基于快速傅里叶变换的方法是精确的，但由于舍入误差，容易出现误差随时间步数线性增加的情况。我们展示了如何修改库利 - 图基算法，以获得与时间步数的平方根成比例的误差增长。最后，对于一个简单模型，我们表明如果将平流求解器用作分裂格式的一部分，我们的结论仍然成立。
