一种“准拉格朗日法”和“欧拉法”统一算法时间积分方案

A new quasi-Lagrangian time integration scheme with the interpolation of fitting bicubic surface

中文关键词:  双三次曲面拟合,准拉格朗日时间积分方案,非线性“三次”大气运动,柯-弗-列判据,双三次曲面平滑

英文关键词:Fitting bicubic surface, Quasi-Lagrangian time integration scheme, Non-linear cubic atmospheric motion, CFL computational stability criterion, Bicubic surface smooth

中文摘要:

      从欧拉算符出发，用泰勒级数展开，给出二阶时空微商余项预报方程,进而讨论一种数值分析新算法——双三次曲面拟合（插值）的准拉格朗日时间积分方案与数值模式。它是将大气运动描述成为非线性的“三次”运动，即是通过对原始大气运动方程中包括标量、矢量的压、温、湿、风、以及“旋转地球上广义牛顿力”加速度场和散度场等，做双三次曲面拟合，实现对各个大气运动变量场的二阶可导，即限定气块上游点在各个不同双三次曲面（片）、具有斜率、曲率和挠率的非线性“三次”变量场上活动，从而可对各个大气运动方程做时间离散积分，即为“双三次曲面拟合—时间步积分—双三次曲面拟合……”，实现成为一种新动力框架数值模式。由于双三次曲面具有数学定律“收敛性”和二阶可导“最优性”，故选用双三次曲面插值求算二阶余差“上游点”，具有“充分必要”的数学理由：它包含了大气运动变量场之斜率、曲率和挠率。因此，埃尔米特双三次曲面片具有对“网格”变量场二阶可导运算“等价性”、及其数学“收敛性”与最佳曲率“最优性”，并且将“准拉格朗日法”与“欧拉法”，以及柯朗 弗里德里希斯 列维判据统一起来。容易实现全球“网格”变量场的双三次曲面拟合，和可按双三次曲面变量场的斜率、曲率或挠率判断，作变量场局域或单点平滑，以此保持“三次”模式的时间积分稳定性。

英文摘要:

      Starting from the Eulerian operator, the forecasting equations with 2-order time and space differential remainder are derived in light of the Taylor series expansion.The quasi-Lagrangian integration scheme with an algorithm of numerical analysis the-bicubic surface interpolation-for a new meteorological numerical model is then suggested. It describes atmospheric motions just as non-linear cubic waves, with fitting different bicubic surfaces to physical scalar or vector variable fields in the model atmosphere, i.e. the pressure, temperature, humidity, wind and divergence as well as the generalized Newtonian force acting to unit air mass on the rotating earth (acceleration). With the procedure of “fitting bicubic surface-time step integration-fitting bicubic surface……” through one to one correspondecnce with the Hermite bicubic patches to parameterize the latitude longitudinal meshes in light of rectangular topologies so as to get the second order derivative, a new model’s dynamic core comes true. Because of that the bicubic surface has two numerical analysis laws of the convergence and the optimality of the second order derivative, it is necessary and sufficient reason mathematically to select the bicubic surface interpolation for calculating the upstream point of a quasi-Lagrangian air parcel. It covers the slope, curvature and torsion of every non linear variable in the atmospheric motions, that is, one Hermite bicubic patch is mathematically equivalent to one secondary derivative (bicubic) variable “mesh” in the model atmosphere, and the former is “convergent” to the latter-with “optimality”. The quasi Lagrangian integration of calculating the upstream point with interpolation on the Hermite bicubic patch is also equivalent to the Eulerian integration with the unitive CFL computational stability criterion. It’s easy to fit a holoscopic variable field to a global bicubic surface, and make some reasonable local area or single point smoothing, according to some conditions of its slope, curvature or torsion, which must conform to the physical constraints such as mass conservation for pressure smooth, energy conservation for temperature smooth, and momentum conservation for wind smooth. Obviously it’s easy to put a new “patch” of bicubic surface on the smoothed place.