全球三次样条格式数值模式动力框架与理想场试验研究

In numerical analysis, cubic spline function consists of cubic spline, bi-cubic surface and tri-cubic (3-D cubic) cube, three based on the cubic spline interpolation (spline scheme), which posses the 2-order differentiable "convergence" and "optimality" of mathematical law, that are: 1) the cubic spline function, together with its first-order and second-order derivatives, contracts to the original function (contraction law); 2) its second-order derivative is optimal approximation to that of the original function (optimality law); 3) there exists periodic cubic spline. We are able to introduce the cubic spline functions into study of a numerical model, which will be a new method or new selection of basic research for innovation of meteorological numerical model (so called "spline model"). The research including: design of global reduced latitude-longitude rectangular mesh to treat both polar areas and Poles; spatial and temporal discretization, derivation and smooth with the spline scheme to variables in the meteorological primitive equations; quasi-Lagrangian time-split integration scheme of solving 3-D paths of all of the upstream points as well as their divergence field with fitting the cubic spline functions; bring about a new dynamic core of a global spline- scheme numerical model on the reduced latitude-longitude grid, with which a full set of exact tests must be experimented on the balance flow, cross-polar flow, Rossby-Haurwitz wave flow and long-time forced simulation, to verify how the (non-topography, hydrostatic) dynamic core being scientific, accurate and its program correctness, try its different dynamical formulations and conservations and check how about the mathematical laws of 2-order differentiable "consistency", "convergence" and "optimality" of the cubic spline functions to use in spline model; and studying on scalable distributed parallel computations to it, to make a more high-efficiency parallel algorithm software to the serial algorithm one of the global spline model. So our research project is to develop an original and innovative dynamic core to a global spline model with the intellectual right proprietor..Global spline/ finite-difference model can be the developing direction in the 21st century and we believe that spline model could prevail against spectral model.

本项目将具有对于原函数及其一阶、二阶导数"收敛性"和对于原函数二阶导数最佳逼近"最优性"等数学性质的三次样条函数引入到数值模式研究中，是对数值模式新方法、新选择的一种探索性创新基础研究。研究内容包括：全球精简（拓扑矩形）经纬网格设计与极区、极点处理；原始大气运动方程变量场三次样条格式二阶时空离散、求导与平滑方案；（二阶可导）三次样条插值求解上游点与三维散度场的准拉格朗日"时间分离+时间分片"积分方案；实现全球三次样条格式（格点）数值模式（无地形、静力）动力框架，对其进行平衡流试验、过极地气流试验、Rossby-Haurwitz波试验、长期积分强迫试验等一系列理想场试验，验证其科学性、精确性及程序正确性；并对其进行可扩展并行算法研究，提交全球三次样条格式数值模式动力框架串行算法和高效并行算法软件，为完成一个具有知识产权与发展前景的全球三次样条格式数值模式进行原创基础研究。

将具有对于原函数及其一阶、二阶导数“收敛性”和对于原函数二阶导数最佳逼近“最优性”等数学性质的三次样条函数引入到数值模式应用中。初步完成准均匀经纬网格三次样条函数（样条格式）变换显式-准拉格朗日积分方案（explicit quasi-Lagrangian integration）。完成研究内容：精简经纬网格设计与极区、极点处理方案；大气运动方程变量场样条格式二阶时空离散、求导、（空间）积分、平滑和顶、底层及侧边界设计方案；二阶可导（解析）三次样条函数插值求解上游点（牛顿）位移与三维位移散度场方案并准拉格朗日“时间分离”积分方案（“时间分离”只用于求非静力垂直位移；“时间分片”只用于给出水平位移初值，这对求极区上游点有效）；初步实现全球样条格式（格点）数值模式（无地形、静力）动力框架，完成平衡流试验、过极地气流试验、Rossby-Haurwitz波试验、及长期积分强迫试验，验证该动力框架（数学）精确性、收敛性及程序正确性；并对其进行可扩展并行算法研究，形成全球样条格式数值模式动力框架串行算法和高效并行算法软件，初步完成一个具有全部知识产权与全新发展前景的全球样条格式数值模式（动力框架）。