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。     

A Two-Step Shape-Preserving Advection Scheme

This paper proposes a new two-step non-oscillatory shape-preserving positive definite finite difference advection transport scheme, which merges the advantages of small dispersion error in the simple first-order upstream scheme and small dissipation error in the simple second-order Lax-Wendroff scheme and is completely different from most of present positive definite advection schemes which are based on revising the upstream scheme results. The proposed scheme is much less time consuming than present shape-preserving or non-oscillatory advection transport schemes and produces results which are comparable to the results obtained from the present more complicated schemes. Elementary tests are also presented to examine the behavior of the scheme.     

显式自由表面模型时间分裂格式的一个积分方案

讨论了有关显式自由表面模型的时间分裂格式中的分裂误差。在传统的积分方案中，由于采用蛙跳格式，斜压动量方程具有三层的时间离散结构，但是，正压动量方程在相应的时间层上只有二层的离散结构。因此，正压方程与斜压方程的时间离散结构是不相容的。分析结果表明，这是导致分裂误差的原因。本提出一种能保持时间离散结构相容的正压和斜压方程的时间积分方案。该积分方案的分裂误差很小因此特别适合于需要长期积分的气候研究。该方案在北太平洋的一个自由表面模型中得到了应用。     

Three-Step Difference Scheme for Solving Nonlinear Time-Evolution Partial Differential Equations

In this paper, a special three-step difference scheme is applied to the solution of nonlinear time-evolution equations, whose coefficients are determined according to accuracy constraints, necessary conditions of square conservation, and historical observation information under the linear supposition. As in the linear case, the schemes also have obvious superiority in overall performance in the nonlinear case compared with traditional finite difference schemes, e.g., the leapfrog（LF） scheme and the complete square conservation difference（CSCD） scheme that do not use historical observations in determining their coefficients, and the retrospective time integration（RTI） scheme that does not consider compatibility and square conservation. Ideal numerical experiments using the one-dimensional nonlinear advection equation with an exact solution show that this three-step scheme minimizes its root mean square error（RMSE） during the first 2500 integration steps when no shock waves occur in the exact solution, while the RTI scheme outperforms the LF scheme and CSCD scheme only in the first 1000 steps and then becomes the worst in terms of RMSE up to the 2500th step. It is concluded that reasonable consideration of accuracy, square conservation, and historical observations is also critical for good performance of a finite difference scheme for solving nonlinear equations.     

JFNK方法概述及其在大气全隐式非静力模式中的应用方案

首先介绍了近年来新发展的非线性方程全隐式数值求解的JFNK方法,及其在地球流体力学方面应用计算实例.可看到,无论在计算精度还是计算效率方面,全隐式数值求解远远超过常规的半隐式计算格式.其次,还讨论了JFNK方法在气象非静力模式中应用方案,并提出了用静力假定和半隐式差分格式来构造预条件处理器,变三维求解为二维求解,简化了方程组求解难度.该方案不仅可用于差分模式,也为用譜方法求解非静力模式提供可能.     

Atmosphere-ocean coupling schemes in a one-diemensional climate model

In this paper, the coupling schemes of atmosphere-ocean climate models are discussed with one-dimensional advection equations. The convergence and stability for synchronous and asynchronous schemes are demonstrated and compared. Conclusions inferred from the analysis are given below. The synchronous scheme as well as the asynchronous-implicit scheme in this model are stable for arbitrary integrating time intervals. The asynchronous explicit scheme is unstable under certain conditions, which depend upon advection velocities and heat exchange parameters in the atmosphere and oceans. With both synchronous and asynchronous stable schemes the discrete solutions converge to their unique exact ones. Advections in the atmosphere and ocean accelerate the rate of convergence of the asynchronous-implicit scheme. It is suggusted that the asynchronous-implicit coupling scheme is a stable and efficient method for most climatic simulations.     

Semi-Implicit Scheme for the DWD Lokal-Modell

Summary  The fourth generation of numerical weather prediction (NWP) models is currently under development at the Deutscher Wetterdienst (DWD) consisting of a global grid point model (GME) and limited-area Lokal-Modell (LM). The nonhydrostatic fully compressible LM has been designed to meet high-resol ution regional fore-cast requirements at meso-β and meso-γ scales. The initial LM implementation is based on the NCAR/Penn State MM5 with the addition of a novel generalized terrain-following coordinate and rotated lat-lon grid. A fully 3D semi-implicit time-stepping scheme has been implemented by retaining the full buoyancy term instead of the approximate form found in MM5. In contrast with earlier schemes, mass-lumping is not applied to simplify the elliptic operator on an Arakawa-C/Lorenz grid. The resulting variable-coefficient elliptic problem is solved using a minimal residual Krylov iterative method with line relaxation preconditione rs. The new semi-implicit scheme is compared with a variant of the Klemp–Wilhelmson split-explicit scheme (horizontal explicit, vertical implicit) on the basis of computational efficiency and accuracy at resolutions ranging from 7 km to 400 m. Both idealized 3D mountain wave flows and naturally occuring flows are analyzed. Below the tropopause, the 3D semi-implicit scheme can be more efficient for low Mach number M ≪ 1 flows when the number of small time steps Δt _s of the split-explicit approach increases with the sound-speed Courant number. Revised October 7, 1999     

包含正定水汽高精度算法的MM5模式的暴雨实例试验

本文首先对非静力中尺度模式ＭＭ５的水汽输送方案作了半理想试验。结果显示,ＭＭ５原有的水汽平 流输送算法在水汽梯度较大的地方产生水汽负值,影响模式的模拟结果和模式的稳定性,对此,本文把一个正定的水汽输送算法Ｐｒａｔｈｅｒ算法引入ＭＭ５,并用实例作了数值试验,检验了模式的模拟能力,结果表明,对ＭＭ５的改进是敏感的有效的,Ｐｒａｔｈｅｒ格式增强了模式暴雨落区,暴雨中心的模拟能力。     

Integration of the shallow water equations in a plane geometry using semi-Lagrangian and Eulerian schemes

Summary Extended integrations of semi-Lagrangian and Eulerian shallow water primitive equation models are performed. The semi-Lagrangian model used the semi-implicit two-time-level scheme. The Eulerian model used a conserving nonlinear advection scheme.For low resolution and longer integrations, difficulties were encountered with the semi-Lagrangian model which were absent in the Eulerian model. These difficulties are discussed.With 14 Figures     

Atmosphere-Ocean Coupling Schemes in a One-Dimensional Climate Model

In this paper, the coupling schemes of atmosphere-ocean climate models are discussed with one-dimensional advection equations. The convergence and stability for synchronous and asynchronous schemes are demonstrated and compared.Conclusions inferred from the analysis are given below. The synchronous scheme as well as the asynchronous-implicit scheme in this model are stable for arbitrary integrating time intervals. The asynchronous explicit scheme is unstable under certain conditions, which depend upon advection velocities and heat exchange parameters in the atmosphere and oceans. With both synchronous and asynchronous stable schemes the discrete solutions converge to their unique exact ones. Advections in the atmosphere and ocean accelerate the rate of convergence of the asynchronous-implicit scheme. It is suggusted that the asynchronous-implicit coupling scheme is a stable and efficient method for most climatic simulations.     

Application of the Characteristic CIP Method to a Shallow Water Model on the Sphere

Semi-implicit algorithms are popularly used to deal with the gravitational term in numerical models. In this paper, we adopt the method of characteristics to compute the solutions for gravity waves on a sphere directly using a semi-Lagrangian advection scheme instead of the semi-implicit method in a shallow water model, to avoid expensive matrix inversions. Adoption of the semi-Lagrangian scheme renders the numerical model always stable for any Courant number, and which saves CPU time. To illustrate the efficiency of the characteristic constrained interpolation profile (CIP) method, some numerical results are shown for idealized test cases on a sphere in the Yin-Yang grid system.     

A comparison of high-order explicit and non-oscillatory finite difference advection schemes for climate and weather models

Summary The Euler equations govern the behavior of a fluid in motion. They have long been used as a test-bed for assessing the accuracy and efficiency of numerical schemes for solving them. This study focused on advection dominated flows so all other terms in the equations are omitted. The study examined only explicit schemes and does not address the many alternative approaches such as semi-Lagrangian and implicit formulations. The schemes examined here were applied to test cases of increasing complexity as it is well-known that a particular scheme may work very well on some test problems but fail on others. To avoid such problems, we began with the standard tests such as the advection of sine waves and various other one-dimensional shapes, then moved on to two-dimensional problems and finally tested cases that have many scales.Summarizing, the results from the range of test cases, it was found that the schemes that perform best are high-order upwind schemes, with diminishing returns above fifth- and sixth-order.Finally, the schemes were applied to cases where scalar advection requires that constraints such as positive-definiteness be satisfied. For these problems, the Flux Corrected Transport (FCT) and weighted essentially non-oscillatory (WENO) schemes were also applied. Similar conclusions were drawn, namely, that fifth- and sixth-order FCT and WENO schemes produced excellent simulations. However, these schemes were more computationally expensive than the standard high-order upwind and centred schemes.     

An efficient non-hydrostatic forecast model

Summary A semi-implicit non-hydrostatic mesoscale forescasting model, which is used operationally in the United Kingdom Meteorological Office, has been reformulated using two time levels and semi-Lagrangian advection so as to permit longer timesteps. The boundaries are found to be an important influence on the model's stability and two methods of suppressing wave reflection are described with the aid of a one-dimensional experiment. The model's responses to two dimensional flow over a ridge and to three-dimensional flow over an isolated mountain are then demonstrated. The reformulated model is more efficient than the original provided a timestep at least twice as long can be used. It is also superior both in having minimal damping (except at the boundaries), and in the accuracy of the advection scheme.     

基于新型LE水平跳点网格的阴阳网格平流方案设计

为了模拟球面平流传输过程,本文基于球面阴阳重叠网格设计了一种两时间层半拉格朗日平流方案.该方案在球面坐标下采用新型的LE水平跳点网格,同时针对阴阳网格重叠区,采用了不同插值方法进行比较分析,且进行了相关的理想数值试验对方案设计效果进行评估.数值试验表明方案设计是成功的,阴阳网格重叠区平流对插值方案比较敏感;半拉格朗日方案能较好地模拟球面刚体平流和变形涡旋的结构、位置及演变过程,并具有较好的数值稳定性和较高的数值精度.     

ENSEMBLE PREDICTION EXPERIMENTS OF TYPHOON TRACK BASED ON THE STOCHASTIC TOTAL TENDENCY PERTURBATION

The GRAPES-TCM is used to make ensemble prediction experiments for typhoon track. Three kinds of ensemble schemes are designed for the experiments. A total of 109 experiments are made for the nine typhoons in 2011 and the integral time is 72 h. The experiment results are shown as follows. In the three ensemble schemes, on the whole, scheme 1 has the best track prediction. Its average absolute track error and overall deviations of typhoon moving speed and moving direction are all the smallest in the three schemes. For both scheme 1 and scheme 2, they are all smaller than those of their control predictions. Both of their ensemble predictions show superiority to their deterministic predictions. Overall, compared with the observations, the typhoon moving directions of the three schemes mainly skew to the right, and in the late integration they mainly tend to be relatively slow. In the three schemes, the track dispersion of scheme 1 is the largest and that of scheme 3 the smallest. In scheme 1 it is much larger than in schemes 2 and 3. The difference of dispersion between scheme 2 and scheme 3 is small. The track dispersions of the three schemes are all much smaller than their rational dispersions. Compared with the eight domestic and overseas operational numerical weather prediction (NWP) models, scheme 1 has better predictions than the other seven operational models except ECMWF NWP model. Scheme 1 has the value of operational application.     

Split-explicit integration of primitive equation barotropic model for the prediction of movement of monsoon depression

The split-explicit version of a limited area primitive equation barotropic model is formulated and tested for the prediction of movement of monsoon depressions. The model is integrated upto 48 hours with split-explicit time inte-gration scheme (Gadd, 1978a) using input of four synoptic cases. The model is also integrated explicitly. The forecast results obtained from both the versions are compared and discussed. The computational time in former version is less than half of the computational time needed in explicit version.     

不同边界层参数化方案对江苏地区一次平流雾过程的模拟影响

边界层参数化方案的选取在平流雾的预报准确度上起着决定性的作用。本文利用WRF模式对2013年3月18~19日发生在江苏地区的一次平流雾过程进行数值模拟试验,对耦合不同闭合方式边界层参数化方案的试验结果与实测气象数据进行对比分析,评估了他们对此次平流雾的模拟效果,探讨了边界层高度对此次平流雾的生成和发展的影响。研究结果表明:（1）耦合不同边界层方案的WRF模式对地面气象要素的模拟结果均呈现气温偏低、湿度和风速偏大的特征。（2）QNSE方案对气温的模拟能力最强;ACM2方案对相对湿度的模拟性能最好;YSU方案对风速的模拟效果最佳。不同边界层方案的模拟结果在垂直方向上的差别主要表现在低空相对湿度上:QNSE方案预报的湿度更大。（3）综合TS（Threat Score）和BS（Bias Score）两个评分指标来看,ACM2方案对雾区分布的模拟效果最好。三个边界层方案对此次平流雾的模拟结果在江苏沿海站点的预报评分较高,在距海较远站点的预报评分表现较差。YSU方案对东南沿海地区的雾区预报评分较高;QNSE方案对长江沿江区域的雾区预报评分较高;ACM2方案对沿海地区、尤其对沿海北部地区的有较好的预报效果。（4）QNSE方案对此次平流雾的生成时间、出现地点预报比较准确。（5）平流雾的生成与发展阶段模拟雾区覆盖范围与边界层高度关系十分紧密,适当强度的湍流混合作用有助于平流雾在地面的生成与发展;但是过强的湍流混合作用会导致大雾过早的消散。     

大地形对Rossby波波射线的影响

本文利用Ｒｕｎｇｅ－Ｋｕｔｔａ方法计算了大地形对定常、非定常低频Ｒｏｓｓｂｙ波波射线的影响，计算结果显示，地形对Ｒｏｓｓｂｙ波波射线有重要影响。对位于地形上游的波源，定常波波射线在传播过程中，遇到地形时，会绕过地形偏向高纬度传播并加密波射线，地形起阻挡屏障作用；波源位于地形中心时，东西走向的地形使波射线发生分支现象，南北走向地形没有这种现象产生。文中还计算了地形存在下，周期为５０ｄ的低频Ｒｏｓｓｂｙ波波射线。结果显示，地形对低频波波射线的影响似乎更复杂。     

高精度迎风偏斜格式的比较与分析

利用一种具有任意阶精度的一般显式有限差分公式构造出高精度迎风偏斜格式，并利用 Fourier分析法评估了这些迎风偏斜格式的耗散误差与频散误差。结果表明，偶阶精度格式的数值相速度快于实际相速度，而奇阶精度格式的数值相速度慢于实际相速度。并且，偶阶精度格式的耗散误差与频散误差低于相邻的奇阶精度格式。为了检验这些格式的计算性能，在一维问题上进行了应用。首先，考虑恒定风场条件下的一维平流试验。主要选择两种不同的初始条件来评价数值格式的精度，这两种试验问题分别是高斯函数、方波函数。试验结果表明，随着数值格式精度的提高，数值格式的误差逐渐减小。而对于高于六阶精度的格式来说，改进的程度并不是很大。其次，应用各阶格式到具有两种不同初始条件的无粘Burgers方程。数值结果表明，随着数值格式阶数的增加，数值结果也得到了明显改进。而对于高于六阶精度的格式来说，进一步的变化并不明显。总之，在兼顾效率与精度条件下六阶迎风偏斜格式是最好的。