偏微分方程保结构算法及其在正压大气浅水波方程上的应用

本项目以高效、稳定、精确地数值求解正压大气浅水波方程为目标，研究和发展偏微分方程保结构算法的基本理论和应用理论。研究内容主要包含两个部分，第一部分是发展保结构算法的基本理论，包括各种构造方法和应用基础，着重考虑差分方法和拟谱方法。第二部分是研究正压大气浅水波方程组的各种保结构算法。首先全面考察该方程组的各种数学结构以及边界条件对这些结构的影响。然后基于这些数学结构，我们重点考虑三类保结构算法，分别是保辛几何结构的差分算法、各种保局部结构的差分算法、保结构的谱和拟谱方法。通过详细的数值试验和比较，本项目不仅会进一步完善保结构算法的基本理论,还将给出一套完整的有关正压大气浅水波方程的数学结构和保结构算法的研究报告。该报告将为系统地把保结构算法应用于计算地球流体力学的研究提供一般的理论基础，也将为更高效、稳定、精确的大气数值模拟提供方法和参考意见。

With the aim to solve the baraotropic atmosphere shallow water equations effeciently, stably and accurately, this project investigates and develops basic theory and applications of the structure-preserving algorithms for partial differential equations. The content of the project has two main parts. In the first part, we develop the basic theory of struture-preserving algorithms with emphesis on the construction methods and applications of finit difference method and pseudo-spectral method. In the second part, we study the structure-preserving algorithms for the baraotropic atmosphere shallow water equations. We first investigate the mathematical structrues of the equations and find out the influence of boundary conditions on those structrues. Then based on the structures, we will consider three kinds of numerical algorithms. They are finite difference methods to preserve Hamiltonian nature, local structure-preserving finite differnece methods and structure-preserving spectral or pseudo-spectral methods. Based on extensive numerical tests and comparisons, this preject will propose a detail report on structures and structure-preserving algorithms of the equations, which not only establishes the basic theory about applications of structure-preserving methods to computational geophysical fluid dynamics, but also proposes some alternative methods and suggestions for efficient, stable and accurate simulations of global atmosphere.