新型有限体积元方法及其在沿海洪水预报中的应用研究

Accurate simulation and prediction of storm-induced coastal flooding is essential for the protection of life and property during the passages of hurricanes. Inaccurate projections of flood areas can either lead to unnecessary evacuation or unexpected loss of life or property. The finite volume element method has been one of the most commonly used numerical methods for simulating the fluid flow in practice. One main attractive property of finite volume element method is that main physical conservation laws possessed in a given application are naturally preserved in the method. Besides, the finite volume element method can be used to deal with domains with complex geometries. This project will propose to design new finite volume element method to simulate coastal flooding. The immersed interface method will be combined with finite volume element method to design immersed interface finite volume element method. By constructing an interpolating function of solution, we are able to give a rigorous error analysis. These will provide the basis for the further development of new finite volume element method and lay a solid mathematical foundation for the development of a new generation of coastal flooding numerical forecasting techniques.

在飓风期间，风暴引起的沿海洪水精确模拟及预测对保护人民生命和财产安全是至关重要的。对洪水地区的不精确预测会造成不必要的人员撤离或导致意外的生命和财产损失。有限体积元方法已成为实际中最常用的模拟流体运动的数值方法之一。有限体积元方法主要的吸引我们的性质是能够自然保持所考察系统的主要物理守恒律。另外，有限体积元方法能够处理复杂几何区域。本项目将设计新型的有限体积元方法来模拟沿海洪水的发展。把浸入面方法与有限体积元方法结合在一起，设计浸入面有限体积元方法。通过构造解的插值函数，我们能够给出严格的误差分析。这些将会为进一步发展新型有限体积元方法提供依据，为研制新一代沿海洪水数值预报技术奠定坚实的数学基础。

沿海洪水精确模拟及预测对保护人民生命和财产安全是至关重要的。本项目针对洪水预报模型设计出新型的有限体积元格式。从描述洪水预报的偏微分方程出发，推导出其在界面上满足的跳跃条件，并根据跳跃条件构造试探范数空间及检验函数空间，完成洪水预报模型的浸入面有限体积元格式的设计工作。进一步，针对建立的浸入面有限体积元格式，利用插值理论及变分形式给出相关的误差估计。同时，我们用新型有限体积元方法研究了随机基流下扰动的线性和非线性发展，结果表明，大的随机震荡会使切线性模式失效。这些结果将会促进有限体积元方法的研究，并为提高沿海洪水的预报精度奠定坚实的数学基础。

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