Comparison and Analysis of Two Different Calculation Schemes of SLEVE- Hybrid Coordinate in GRAPES_MESO Model

The calculation scheme of the smoothed-level and hybrid (SLEVE-hybrid for short) coordinates in numerical forecasting model is not limited to one. It is divided into the semi-analytical scheme and the finite differential scheme in terms of the various differential methods of the coordinate deformation variables. Comparing the dynamic equation and the long-time batch simulation results of the two schemes, the present study draws the following conclusions. The first- order finite difference accuracy of the coordinate deformation variables in the finite differential scheme is theoretically lower than that in the semi-analytical scheme. The larger the vertical gradient of the layer thickness is, the larger the relative errors of the finite differential scheme are. The long-time batch simulation test in the GRAPES model dynamic core demonstrates that the bias of the temperature and the geopotential height in the semi-analytical scheme is smaller under the default layering, while the simulation difference of the two schemes is greatly reduced when the layering is more uniform.     

Comparison and Analysis of Two Different Calculation Schemes of SLEVE-Hybrid Coordinate in the GRAPES＿MESO Model

The calculation scheme of the smoothed-level and hybrid(SLEVE-hybrid for short) coordinates in numerical forecasting model is not limited in number. It is divided into the semi-analytical scheme and the finite differential scheme in terms of the various differential methods of the coordinate deformation variables. Having compared the dynamic equation and the long-time batch simulation results of the two schemes, the present study draws the following conclusions. The first-order finite difference accuracy of the coordinate deformation variables in the finite differential scheme is theoretically lower than that in the semi-analytical scheme. The larger the vertical gradient of the layer thickness is, the larger the relative errors of the finite differential scheme are. The long-time batch simulation test in the GRAPES model dynamic core demonstrates that the bias of the temperature and the geopotential height in the semianalytical scheme is smaller under the default layering, while the simulation difference of the two schemes is greatly reduced when the layering is more uniform.     

最优差分方案

在数值预报和数值模拟中,描述空间微分项的最主要的方法是有限差分法,但使用差分方法会引入截断误差.伍荣生1979年指出,通过在原物理场的基础上构造一个新的物理场,替代原物理场进行差分计算,可以达到减小误差的目的.该文是伍荣生1979年工作的继续,目的在于解释伍荣生1979年所构造的差分格式并得到更为一般化的差分格式.文中给出新的差分格式结合了经典有限差分方法的快速计算和谱方法的高精度的优点.如果在一个给定的网格上对气象要素场进行离散傅里叶级数展开,则基函数(正弦或余弦)的频谱是事免已知的.作者将伍荣生1979年构造物理场的方法视为对物理场的一次平滑,探讨了获取二次平滑场、多次平滑的一般化方法.获取平滑场的基奉原理是使得在固定频谱上的差分逼近程度达到最优.通过对频谱上的累计误差的下降速度分析表明,平滑次数的上限为3次.数值分析的结果表明,二次平滑的最大误差是未作任何平滑的最大误差的0.04倍,在使用相同计算代价的情况下,二次平滑的最大误差是经典的差分格式的0.3倍.平流试验的结果也表明,新的差分格式即一次平滑、二次平滑方案的结果远远优于经典的差分格式.新的差分格式意义在于,在不加密网格的情况下提供了一条提高数值计算精度的途径.     

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.     

Generalized Square Conservative Multistep Finite Difference Scheme Incorporating Historical Observations

In this paper,a multistep finite difference scheme has been proposed,whose coefficients are determined taking into consideration compatibility and generalized quadratic conservation,as well as incorporating historical observation data.The schemes have three advantages:high-order accuracy in time,generalized square conservation,and smart use of historical observations.Numerical tests based on the one-dimensional linear advection equations suggest that reasonable consideration of accuracy,square conservation,and inclusion of historical observations is critical for good performance of a finite difference scheme.     

A NUMERICAL SIMULATION OF THE GEOSTROPHIC ADJUSTMENT PROCESS

In this paper,a numerical simulation of the geostrophic adjustment process with C-grid network is illustrated.A difference scheme which has the energy and potential vorticity conserving relation consistent with the differential equations is given,and the effect of some time difference schemes on dispersion of the gravity-inertia wave is discussed.An improved forward-backward time integration scheme is proposed for keeping the computational stability.The effect of various boundary conditions for a finite region model On the gravity-inertia wave is shown by some calculated results.     

Application of a Shape-Preserving Advection Scheme to the Moisture Equation in an E-grid Regional Forecast Model

This paper presents a methodology which is very useful to design shape-preserving advection finite difference scheme on general E-grid horizontal arrangement of variables through introducing a two-step shape-preserving positive definite advection scheme in the moisture equation of the LASG-REM (LASG regional E-grid eta-coordinate forecast model). By trial-forecasting six local heavy raincases, the efficiency of the shape-preserving advection scheme in practical application has been examined. The LASG-REM with the shape-preserving advection scheme has a good forecasting ability for local precipitation.     

一类计算稳定性好的显式平流差分格式

通常的显式平流差分格式，如迎风格式，Lax-Wendroff格式等，均是有条件稳定的,其稳定条件与差分网格的时、空步长有关。本文对线性和拟线性平流方程分别构造了一种计算稳定性好的显式差分格式。对前一格式，本文严格证明了它的无条件稳定性及收敛性，并具有一阶精度；对后一格式，由于非线性方程的限制，本文用数值试验研究了它的计算稳定性。     

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.