三阶龙格-库塔时间分裂显式算法的误差分析

分析了新一代非静力中尺度数值模式中常用的三阶龙格-库塔时间分裂显式算法(RK3)的稳定性和误差性质,特别是分析了空间中央差分和迎风偏斜两种不同情况下该算法不同的稳定性和误差性质。运用数学软件先进的符号计算功能,分析了该算法涉及的复杂高阶、高次幂振幅矩阵的特征值性质;并通过一维线性声波-平流方程组的数值模拟实验,检验了时间分裂算法的模拟效果。对振幅矩阵特征值模的表达式进行高阶的级数展开,得到了该算法的分裂误差项的公式;而且,由于特征值模的公式保留了较高阶项,可以同时分析迎风偏斜和中央差两种空间差分格式的分裂误差性质。根据分裂误差项公式,定量地比较了三阶和二阶龙格-库塔格式(RK2)的分裂误差大小以及误差与小时间步数的关系,发现迎风格式RK3的分裂误差明显小于RK2的误差,并具有更好的稳定性质。空间中央差格式的分裂误差项具有更高阶数,比迎风格式具有更小的时间分裂误差。对于各种不同波长的特征值分析和采用中央差格式的数值模拟,也进一步证实空间差分采用中央差时,RK3时间分裂显式算法在不同方向传播的声波振幅几乎没有差别。另外,误差公式以及数值试验结果说明RK3的分裂误差也略小于Adams-Bashforth-Moulton分裂显式法的分裂误差。

The error analysis of the third order Runge Kutta time split explicit scheme.

Third order Runge-Kutta time split-explicit scheme is generally used in non-hydrostatic atmospheric models.The stability and error of this scheme is analyzed in this paper.The amplification matrix of the scheme is of high order and high power.The derivation of its eigenvalues formula is performed with the Wolfram Mathematica software which handles complex symbolic calculations and a numerical test of one-dimensional linearized sound-advection equations is performed to show the performance of this scheme.The Taylor series expansion of the norms of the eigenvalues is performed with higher-order terms kept and the splitting error term is derived out for both upwind and centered spatial differences.The splitting error and its relationship with the number of small time steps are shown directly and quantitatively.It is demonstrated that the scheme is numerically stable and has a smaller splitting error than second-order Runge-Kutta time split-explicit schemes.The splitting error term of centered spatial differences has higher order and so the split error is smaller than that in upwind schemes.These conclusions are also demonstrated by the analysis of the eigenvalues of propagating sound waves with different wave-length and the sound wave advection numerical experiment for the second-order centered spatial difference with the RK3 split-explicit scheme for which there are almost no differences in the forward and the backward moving mode.Moreover,The splitting error of the RK3 split-explicit scheme is also smaller than the Adams-Bashforth-Moulton scheme.