基于高维散乱数据的高精度拟插值格式的构造及其在双曲方程求解中的应用

The interpolation method determines the coefficients of the interpolation function by solving a large linear systems whose interpolation matric might be ill-conditioned. Consequently, not only the computation expense is very large but also the process is unstable. The quasi-interpolation method gives the expression of approximation function directly, does not require solution of any linear system and hence is gradually becoming an important approximation approach to a large set of scattered data. The quasi-interpolation scheme based on scattered data has more extensive application value than one based on uniform grids, but then the study on the multivariate quasi-interpolation method based on scattered data is now being placed on start in. The project plans to develop the work of innovation in the construction of high-accuracy quasi-interpolation schemes based on high-dimension scattered data and its application to the solution of hyperbolic equations. The project particularly study the constructions and the approximation error analysis of high-accuracy multivariate gobal quasi-interpolation schemes based on scattered data and high-accuracy multivariate local quasi-interpolation schemes based on scattered data, and use these quasi-interpolation schemes to solve hyperbolic equations numerically, so as to attempt to develop a solution scheme of hyperbolic equations with uniform form for mixed mesh. For the purpose, the project plans to solve the following problems: the construction of basic multivariate quasi-interpolation function with the reproduction of few degree polynomial and shade-preserving,the construction of weighted functions with specific properties, the construction of high-order numerical differential formula on finite set of scattered data and the addition of artificial viscosity. The aim of the project is to supplement the technique and theory of quasi-interpolation approximation method and generalize its application.

插值法在逼近散乱数据时需要通过求解大规模的线性方程组来确定插值系数，此时不仅计算规模大且不稳定。拟插值方法直接给出逼近函数式，避免了求解大规模线性代数方程组，正成为大规模数据逼近的重要工具。基于散乱数据的拟插值方法比基于网格点集的拟插值方法有着更广泛的应用价值，但基于散乱数据的多元拟插值的构造工作目前处于起步之中。本项目拟在基于高维散乱数据的高精度拟插值格式的构造及其在双曲方程求解中的应用开展创新性研究工作，重点研究基于散乱数据的高精度多元全局和局部两种形式的拟插值格式的构造及其逼近分析，并将其用于双曲方程的数值求解，以尝试发展一种能适应于混合网格且具有统一形式的双曲方程求解格式。为此拟解决以下关键问题：具有低次多项式恢复和保形特性的基本型多元拟插值格式的构造；具有某些特性的权函数的构造；有限散乱点集上的高阶精度的数值微分公式；人工粘性的添加方法。目的在于完善拟插值方法与理论，推广其应用。

高维散乱数据广泛存在于工程技术和科学计算等领域，常常被需要解释和利用。拟插值是散乱数据函数逼近的一种有效方法，目前的方法多针对二维散乱数据，高维下的拟插值函数的构造正处于发展中。. 主要研究内容：二维矩阵格点型数据高精度逼近的拟插值格式的构造，三维散乱数据高精度逼近的拟插值格式的构造，任意维散乱数据高精度逼近的拟插值格式的构造以及这些所构拟插值格式的逼近误差分析；利用所构拟插值构造解双曲方程的求解格式；基于径向基函数插值的高精度差商公式的构造及其在解微分方程中的应用。. 重要结果：构造了一组适用于二维矩阵格点型数据逼近的具有任意次代数精度多元multiquadric拟插值算子；在二维散乱结点集的三角剖分上分别利用构造的具有任意次代数精度的修正的平均值坐标插值的线性组合和三角形上的Hermite插值构造了逼近三维散乱数据的高精度分片光滑的有理拟插值格式和分片连续的Hermite插值格式；将此类修正的平均值坐标插值移值到以待估值点在散乱结点集中的自然邻点为顶点的多边形上，从而构造了一个不需要固定网格的逼近三维散乱数据的有理拟插值格式；将自然邻点的概念与具有任意次代数精度的局部径向基函数插值结合在一起构造了一类适合于任意维散乱数据逼近的局部拟插值格式。对以上所构全部拟插值格式的逼近误差进行了分析。. 利用插值重构和Lax-Wendroff格式构造了一类在时空具有三阶逼近精度且满足TVD性质的求解双曲方程的简单格式，且在特殊参数选择下可达到四阶精度；利用具有零次代数精度的径向基函数插值的Lagrange形式给出了逼近被插函数导数的是同节点模板的多项式差商逼近阶两倍的高阶RBF差商，并利用这些RBF差商分别高效地求解了常微分方程和Poisson方程等边值问题。. 科学意义：本项目所构系列拟插值可分别适合不同类型的散乱数据逼近，且这些拟插值格式结构简单，易于实现，执行效率高，特别适合于大规模散乱数据逼近。所构造的具有最佳参数的RBF差商开创了一种构造解微分方程的紧致超收敛差分格式的新途径。