Improvement of accuracy and stability in numerically solving hyperbolic equations by IDO (Interpolated Differential Operator) scheme with Runge-Kutta time integration
Improvement of accuracy and stability in numerically solving hyperbolic equations by IDO (Interpolated Differential Operator) scheme with Runge-Kutta time integration
复制标题
通过采用龙格-库塔时间积分的 IDO（插值微分算子）方案提高数值求解双曲方程的精度和稳定性
DOI：
10.1002/ecjc.10127
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
T. Utsumi
中科院分区：
文献类型：
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作者：
H. Yoshida;T. Aoki;T. Utsumi
为了利用插值型微分算子(IDO)格式求解双曲型偏微分方程组，采用泰勒展开对因变量进行时间积分，并用空间微分代替时间微分。然而，在这种方法中，时间精度受到插值函数阶数的限制，另外，由于计算的复杂性，空间精度在多维问题中是不够的。在数值稳定性方面，由CFL(Courant-Friedrich-Levy)数表示的稳定区域较窄。因此，为了提高时空精度和保证数值稳定性，采用了龙格-库塔法的时间积分方法。在此基础上，利用IDO格式以物理量及其空间导数为因变量的特点，提出了一种在不增加计算代价的情况下提高Runge-Kutta方法阶数的方法。对二维平流方程和一维波动方程进行了求解，并与常规泰勒展开式的结果进行了定量比较。此外，为了证明这种方法对实际问题的适应性，我们考虑了Williamson对球面几何中的浅水方程的测试用例5。结果表明，多维龙格-库塔法比泰勒展开法具有更高的精度和稳定性，证明了该方法的有效性。《威利期刊》2003年第3期，87(2)：33-42,2004；在线发表在《威利国际科学》(www.intercience.wiley.com)上。DOI 10.1002/ecjc.10127
In order to solve hyperbolic partial differential equations by means of the Interpolated Differential Operator (IDO) scheme, time integration of the dependent variable has been carried out by Taylor expansion, and time differentiation has been performed by replacing it with a spatial differentiation. However, in such a method, the time accuracy is limited by the order of the interpolation function and in addition the spatial accuracy is not sufficient in multidimensional problems due to the complexity of the calculations. In terms of numerical stability, the stable region indicated by the CFL (Courant-Friedrich-Levy) number is narrow. Hence, in order to improve the space-time accuracy and to secure numerical stability, time integration by the Runge-Kutta method is applied. Further, a method for increasing the order of the Runge-Kutta method without increasing the computational cost is proposed, taking advantage of the characteristics of the IDO scheme with the physical quantity and its spatial derivative as the dependent variables. The two-dimensional advection equation and the one-dimensional wave equation are solved and the results are quantitatively compared with those obtained by the conventional Taylor expansion. Also, in order to demonstrate the adaptability of this approach to practical problems, we consider Williamson's test case 5 for the shallow-water equation in spherical geometry. It is found that the Runge-Kutta method for multiple dimensions yields accuracy and stability higher than those of the Taylor expansion, demonstrating the effectiveness of the approach. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(2): 33–42, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10127