应用于双向套网格模式的一种变格距差分解法

本文提出了一种用于双向套网格模式的变格距差分计算方案。该方案在不同格距的网格区采用不同精度的差分格式,它自然地连接粗细网格,避免了一般套网格方案在粗细网格相重合点上进行的重复计算。用解析法和数值试验证明了：它与其它一些变格距差分格式相比,对短波的穿透能力有明显改进,虚假的反射也较小。应用该方案建立了正压原始方程双向套网格模式,并采用空间分解和时间分解计算方法。这不仅使二维问题转化为二个一维问题,而且二维套网格也可简化为一维均匀网格和一维套网格两部分,从而使计算和程序简化。用理想场为初值所做的一系列数值试验表明,该模式中的波可以自由进出粗细网格区,计算稳定。最后,还用该模式做了台风路径预报试验,给出了一些试验结果。     

半隐式半拉格朗日平方守恒计算格式的构造

在显式半拉格朗日完全平方守恒格式基础上, 构造出半隐式半拉格朗日完全平方守恒计算格式, 它继承了显式半拉格朗日完全平方守恒格式的优点, 并突破计算不稳定柯朗条件对时间步长的约束, 使时间步长大为增长.此外, 还给出这种新的计算格式在一维原始方程上的应用.     

GRAPES区域模式水汽平流方案的比较与改进

与欧拉显式时间差分方法相比,GRAPES区域模式采用半隐半拉格朗日时间差分方案可增加时间步长且不影响稳定性,而且模式积分可有较高的计算效率和准确性。半拉格朗日法需要用到内插算法来预测下一时刻的值,对于水汽场的内插值来说,常常会造成预报值的过饱和或者是负值,需要进行特殊处理。比较GRAPES模式的准单调半拉格朗日方案(QMSL)和高精度正定保形方案(PRM),分析模式的降水预报、形势预报,同时初步总结了两方案的优缺点。在参考LCSL(Linear Constraint Semi-Lagarangain)方案的基础上,改进QMSL方案,通过连续试验运行,表明新方案基本稳定可靠,对于降水预报、形势预报有一定的改进,在台风预报试验中也有良好的表现。     

重力波特征线显式数值解法试验

利用CIP(Constrained Interpolation Profile)特征线半拉格朗日解法对平面一维和二维浅水波方程的重力波和普通经纬度网格下的球面浅水波模式进行了数值求解试验,给出了CIP特征线法的数值结果和对应大时间步长积分的对比。结果显示,一维CIP特征线法对重力波的位相、振幅和传播有很好的描述,对于不同时间步长都显示出计算稳定的特性。空间分离的二维特征线法是多维问题计算中简单有效的方法,在平面和球面二维问题的数值结果中都得到了CIP特征线法稳定的计算和良好的波动传播结果,在浅水波方程中可以克服半隐式算法在高分辨率时的大量运算,并有望成为处理重力波的替代算法。     

初始方程数值模式的一个计算格式在有限区域预报的试验

本文是将我组提出的初始方程数值预报模式的一个计算格式,在有限区域进行预报的试验。 本文主要叙述了侧边界条件试验的情况,初值温度的计算方案,最后给出一次24小时的预报结果。     

一类计算性系统误差消除与斜压原始方程天气气候模式改进

斜压原始方程半隐式全能量守恒格式的构造问题长期没有解决。本研究在成功地构造实现其全能量完全守恒的半隐式方案基础上，进行了此守恒方案与欧洲中期天气预报中心（ECMWF）的σ－坐标原始方程全球谱模式半隐式方案间的实际资料对比实验。实验表明，850hPa平均预报高度场RMS误差在积分一周以后得到明显改进，到第30天其预报误差降低达到了50％，进一步的对比实验表明，对流层中部和下部的月预报平均高度场RMS误差也显降低，而且一些明显的系统性误差也得到大幅度改进。更加详细的分析显示，这些收益的很大一部分是从超长波成分的改进中得到的。这说明，通过构造守恒性时间差分方案消除了响应的计算性系统误差源汇，进而能够使模式气候漂移得到显改进，而这种误差源汇存在于传统的，现仍被普遍采用的斜压原始方程天气气候模式中。     

正压原始方程半隐式样条格式

本文把样条函数应用于半隐式正压原始方程模式,分析了这种格式的精度、线性稳定性条件,讨论了一种用三次样条函数求解Helmholtz方程的迭代法,并证明了迭代的收敛性。实例分析表明,半隐式样条格式计算稳定,预报场光滑完整,与一般差分格式相比,预报系统移速偏慢现象有较明显改进,用于台风路径预报,其结果有参考价值。     

正压原始方程模式采用准拉格朗日格式的预报试验

本文设计了一个时间积分采用准拉格朗日格式的台风路径预报的正压原始方程模式。模式采用海绵型开放边界,使边界附近的预报有所改进。采用海绵型边条件和准拉格朗日平流格式后,模式的计算稳定性较好。积分24小时,区域总能量和涡度拟能变化小于1%。在用实际观测资料所做的台风路径预报试验中,初始风场的散度是作为源项加到连续方程中的。对比试验结果表明:有实际源项的初始场的预报优于无源项和源项被人为改变了的初始场的预报,从而说明起始风场中包括非地转部分有利于改善台风路径预报。     

发展方程计算稳定性的比较分析

对线性与非线性发展方程的几种差分格式,以一维线性和非线性平流方程为例,对线性与非线性发展方程差分格式的计算稳定性进行了比较分析,揭示了差分格式结构和初值形式与非线性计算稳定性的关系。     

正压原始方程套网格试验

本文用“单向”(one—way)方法进行正压原始方程套网格试验。粗、细网格均采用空间分解准拉格朗日格式和由廖洞贤等提出的与此差分格式一致的内边界条件(即方案二)。试验结果表明,用方案二作出的24小时细网格预报,与按半动量格式和粗网格直接给细网格提供内边界条件(即方案一)所得的结果相比,去除了在流出处出现的不稳定,系统的流入、流出也较好。这两种方案再加上Davies“松弛区域”的方法(分别为方案四,方案三),其结果与上述类似,且方案四比方案二还要好些。     

Studies on Non-interpolating Semi-Lagrangian Scheme and Numerical Solution to KdV Equation

A new non-interpolating semi-Lagrangian scheme has been proposed, which can eliminate any interpolation, and consequently numerical smoothing of forecast fields. Here the new scheme is applied to KdV equation and its performance is assessed by comparing the numerical results with those produced by Ritchie’s scheme (1986). The comparison shows that the non-interpolating semi-Lagrangian scheme appears to have efficiency advantages.     

半拉格朗日平方守恒计算格式的研究

作者研究了一种新的显式平方守恒的半拉格朗日计算格式.该格式是欧拉空间平方守恒格式在半拉格朗日空间的推广和发展,并采用了守恒插值方法.此外,还给出这种格式在一维原始方程上的应用.     

Improvement of the semi-Lagrangian advection scheme in the GRAPES model: Theoretical analysis and idealized tests

ABSTRACT The Global/Regional Assimilation and PrEdiction System （GRAPES） is the newgeneration numerical weather predic- tion （NWP） system developed by the China Meteorological Administration. It is a fully compressible non-hydrostatical global/regional unified model that uses a traditional semi-Lagrangian advection scheme with cubic Lagrangian interpola tion （referred to as the SL_CL scheme）. The SL_CL scheme has been used in many operational NWP models, but there are still some deficiencies, such as the damping effects due to the interpolation and the relatively low accuracy. Based on Reich＇s semi-Lagrangian advection scheme （referred to as the R2007 scheme）, the Re_R2007 scheme that uses the low- and high-order B-spline function for interpolation at the departure point, is developed in this paper. One- and two-dimensional idealized tests in the rectangular coordinate system with uniform grid cells were conducted to compare the Re..R2007 scheme and the SL_CL scheme. The numerical results showed that： （1） the damping effects were remarkably reduced with the Re_R2007 scheme; and （2） the normalized errors of the Re_R2007 scheme were about 7.5 and 3 times smaller than those of the SL_CL scheme in one- and two-dimensional tests, respectively, indicating the higher accuracy of the Re..R2007 scheme. Furthermore, two solid-body rotation tests were conducted in the latitude-longitude spherical coordinate system with non uniform grid cells, which also verified the Re_R2007 scheme＇s advantages. Finally, in comparison with other global advection schemes, the Re_R2007 scheme was competitive in terms of accuracy and flow independence. An encouraging possibility for the application of the Re_R2007 scheme to the GRAPES model is provided.     

On the forward-in-time upstream advection scheme for non-uniform and time-dependent flow

Summary The classical forward-in-time upstream advection scheme for uniform flow field has been extended to include non-uniform and time-dependent advective flow. This generalised scheme is described in one dimension for an advective flow which varies both in time and in space. The classical upstream advection scheme is only first-order accurate both in time and in space if the advective flow is not uniform. Higherorder accuracy in both time and space, however, can be easily obtained in the generalised scheme.This generalised scheme with third-order accuracy is applied to the one-dimensional inviscid Burgers equation (socalled self-advection problem), two-dimensional steady flow, and to a time-split shallow water equation model. The results are compared with those obtained from the Takacs' (1985) scheme and from a standard third-order semi-Lagrangian scheme, and also with those obtained from the fourth-order Lax-Wendroff scheme of Crowley (1968) in the time-split shallow water equation model. It is shown that the generalised scheme performs as well as, but is more efficient than, the standard semi-Lagrangian scheme with same order. It is much more accurate than the Takacs' scheme which has large dissipation errors, especially for the flow with strong deformation. In contrast, the generalised scheme has very weak dissipation and has much better dispersion and shapeconserving properties. Although the fourth-order Lax-Wendroff scheme has higher accuracy and can give more accurate numerical solutions for uniform advective flow or solid rotational flow (Crowley, 1968), it is inferior to the generalised third-order scheme for non-uniform flow with strong deformation or large spatial gradients. This generalised scheme, therefore, has considerable application potential in different numerical models, especially for the models using time-split algorithms.With 8 Figures     

Multispectrum Method and the Computation of Vapor Equation

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A monotonic limiter and flux-corrected transport approach for high-order time integration schemes

A monotonic transport algorithm for a high-order time integration scheme is described in this paper. The algorithm is a modified version of an existing high-order time integration scheme, and is tested using a simple one-dimensional pulse and two-dimensional deformational flows. It is found that the new formulation can remove the error, caused by new maxima/minima and excessive smoothing, which occurs in scalar transport using the original extrapolation scheme. The results show that there may be potential for the high-order time integration scheme to be applied in numerical weather prediction models.     

GRAPES模式中高精度正定保形物质平流方案的研究Ⅰ:理论方案设计与理想试验

平流计算的精度对数值模式的结果有着重要影响.如何在半拉格朗日模式中发展高阶精度的标量平流计算方案是提高半拉格朗日数值模式精度的重要问题.文中采用计算流体力学中一个新的高精度正定保形的物质平流方案,通过映射单元格方法将其与半拉格朗日模式结合起来,既保留了半拉格朗日时间积分方案中积分时间步长大、计算效率高的特点,又发挥新方...     

Cascade插值方法在GRAPES模式中的应用

基于半拉格朗日 (semi-Lagrangian) 方案的数值天气预报模式, 求解半拉格朗日轨迹上游点变量, 通常采用传统直线逐点拉格朗日多项式插值, 由已知模式格点 (欧拉网格点) 的数值插值获得。对于三维空间上游点的插值, N阶精度需要O(N3) 运算量。N增大, 运算量将大幅增加, 特别耗费计算机机时, 而采用Cascade插值法 (降阶插值法) 则只需要O(N) 运算量。它的显著特点是:用曲线代替直线, 通过一系列中间过渡网格点, 在曲线上用一维拉格朗日插值, 使得相邻拉格朗日格点或中间过渡点的插值不再是孤立的, 而且可以重复使用某些中间结果, 达到减少运算量的目的。将这种方法合理应用于GRAPES模式, 并根据模式的特点, 对Cascade插值过程中独立变量的距离分段计算, 从而有利于实现并行计算。计算结果表明Cascade插值法与传统直线逐点插值法相比, 计算效率平均提高约30%, 同时不降低精度。     

An improved dynamic core for a non-hydrostatic model system on the Yin-Yang grid

A 3D dynamic core of the non-hydrostatic model GRAPES(Global/Regional Assimilation and Prediction System) is developed on the Yin-Yang grid to address the polar problem and to enhance the computational efficiency. Three-dimensional Coriolis forcing is introduced to the new core, and full representation of the Coriolis forcing makes it straightforward to share code between the Yin and Yang subdomains. Similar to that in the original GRAPES model, a semi-implicit semi-Lagrangian scheme is adopted for temporal integration and advection with additional arrangement for cross-boundary transport. Under a non-centered second-order temporal and spatial discretization, the dry nonhydrostatic frame is summarized as the solution of an elliptical problem. The resulting Helmholtz equation is solved with the Generalized Conjugate Residual solver in cooperation with the classic Schwarz method. Even though the coefficients of the equation are quite different from those in the original model, the computational procedure of the new core is just the same. The bi-cubic Lagrangian interpolation serves to provide Dirichlet-type boundary conditions with data transfer between the subdomains. The dry core is evaluated with several benchmark test cases, and all the tests display reasonable numerical stability and computing performance. Persistency of the balanced flow and development of both the mountain-induced Rossby wave and Rossby–Haurwitz wave confirms the appropriate installation of the 3D Coriolis terms in the semi-implicit semi-Lagrangian dynamic core on the Yin-Yang grid.