径向基函数逼近方法在构造无网格保能量算法中的应用

近几十年来，保结构算法的诞生和发展对科学计算领域具有重大的理论和应用意义，其本质是保持原方程的基本特征不变。保能量算法作为一种常见的保结构算法，相比传统非守恒数值算法具有更高的精度和更好的稳定性。然而，现有的保能量算法大多适用于规则区域上的等距采样数据。为了弥补不足，本项目将研究如何用径向基函数方法构造无网格保能量算法。径向基函数可以有效地处理不规则区域上的多元、散乱数据，已被广泛应用于各个领域。本项目的主要研究内容包括：首先，探索径向基函数拟插值方法对高阶导数的逼近，并研究将其用于构造一维非线性波方程的无网格保能量算法，进一步，希望考虑多维非线性偏微分方程，探索利用径向基函数Galerkin方法来进行空间离散，再结合已有的时间离散方法来获得保能量算法，同时在理论上分析数值解的误差和逼近阶。最后，从球面上的偏微分方程出发，探讨球面径向基函数方法在构造保能量算法中的应用。

The development of structure-preserving methods in the last few decades plays an important role in the field of scientific computing, the nature is to preserve the structure of original equations. Energy-preserving schemes are a type of structure-preserving methods. They are more accurate and stable than non-conservative methods. However, most of existing energy-preserving schemes are only valid for uniform sampling data on regular domain. To cover this shortage, this project will study how to use radial basis function (RBF) methods to construct meshless energy-preserving schemes. RBF method is a powerful tool to handle multi-variate and scattered data on irregular domain. This project will focus on the following three points. Firstly, we will explore the convergence of RBF quasi-interpolation methods when approximating high-order derivatives, and use it to construct meshless energy-preserving schemes for one-dimensional nonlinear wave equations. Secondly, for multi-variate nonlinear equations, we will use RBF Galerkin methods for spatial discretization and employ an appropriate time integrator to obtain energy-preserving schemes. Moreover, the error and convergence between the approximate solution and exact solution will be analyzed in detail. At last, we start with the partial differential equation on sphere, study applications of spherical RBF methods in constructing energy-preserving schemes.