“准拉格朗日法”和“欧拉法”的数学一致性与三次插值函数算法时间积分方案

The Mathematical Consistency of Quasi-Lagrangian and Eulerian Time Integration Schemes with Fitting Cubic Interpolation Function

从预报方程组通式和欧拉算符出发,用泰勒级数展开,推导出时空间微商余项为二阶、四阶完全预报方程组。可以证明,同阶时空间微商余项的准拉格朗日法和向前差分欧拉法时间积分方案,具有相同的物理意义和数学一致性,相比之下,传统中央差欧拉法时间积分方案是"简单格式"与"增大不可预测计算误差"并存。进而讨论用"三次插值函数"实现二阶时空间微商余项准拉格朗日法、或同阶向前差分欧拉法(可"二选一"),它们应该分别替代"双线性插值"准拉格朗日法和传统时空间中央差欧拉法,因前二者时空间微商余项及计算精度高于后二者。所以,三次插值函数算法可将准拉格朗日法和欧拉法时间积分方案、以及CFL判据统一起来。由于三次插值函数具有对原函数"变量场"的数学定律"收敛性"和二阶可导"最优性",且一次"三次插值函数"运算,即具对网格变量场二阶可导拟合"等价性":不仅拟合变量场斜率、还拟合其曲率和挠率。并因周期"三次插值函数",可作全球变量场"三次插值函数"二阶可导拟合,实现全球"三次"数值模式,并且可按变量场曲率判断,作变量场局域或单点平滑,保持"三次"模式时间积分的稳定性。

The Navier-Stokes primitive equations as well as Eulerian operator is discussed,and the perfect forecasting equations with time and space 2-order,4-order differential remainders by Taylor series expansion is derived.It can be proved that,with same space-time order of remainder,the quasi-Lagrangian integration scheme has the same physical meaning and mathematical consistency to the Euler′s.In contrast,the traditional time central difference is ＇simple format＇ and ＇increased the unpredictability of calculation error＇ co-exist.Further discussion of that,with fitting of ＇cubic interpolation function＇,the realization of the quasi-Lagrangian and Eulerian integration schemes（be ＇a choice＇） with 2-order time and space derivative remainder,both should respectively take place the quasi-Lagrangian ＇bilinear interpolation＇ and the Eulerian space central difference,i.e.slope and curvature,because of more accurate calculation and higher time and space order of the remainder in former two than in latter two.Therefore,the cubic interpolation algorithm can be the quasi-Lagrangian and Eulerian integration schemes unified,together with the ＇CFL＇ criterion.Because of the cubic interpolation function with numerical analysis of the law of ＇convergence＇ and the second-order derivative ＇optimality＇,and that one ＇cubic interpolation＇ is equivalent in perform operation to the secondary derivative ＇mesh＇ variables all of forecasting equations,which is fitting not only one variable′s slope,but also its curvature and torsion.And the period ＇cubic interpolation function＇ can be used for global variables by fitting the period cubic spline/bicubic surface/tricubic cube,in order to achieve the global ＇cubic＇ numerical model,as more as,it is according to the curvature or torsion of one cubic variable field,judge the reasonable local area or simple point to be smooth,in order to maintain the model stability.