【作者】 袁美；

【导师】 宋松和；

【作者基本信息】 国防科学技术大学 ， 数学， 2006， 硕士

【摘要】 浅水波方程在海洋工程、环境问题、生态学等许多领域有着广泛的应用,众所周知,浅水波方程的解很难求得,而且即使在初值很光滑的情况下其解也会出现间断。本文主要讨论了非结构网格上解浅水波方程的最小二乘有限体积方法。本文共分五章:第一章简要介绍了浅水波方程的物理背景、数学模型及其数值解法的研究发展。第二章介绍了结构网格上和非结构网格上解浅水波方程的数值方法,详细介绍了具有代表性的求解浅水波方程的Godunov型有限体积格式和复合有限体积格式。第三章介绍了双曲型守恒律有限体积法的框架、网格和控制体积、TVD型Runge-Kutta时间离散方法、极值原理以及限制器形式。第四章构造出了非结构网格上解浅水波方程的非振荡有限体积方法。借助最小二乘的思想,得到了单元上的一个线性插值多项式,并利用极值原理的思想,在保证其解不出现新的极值的情况下,构造出了非结构网格上解浅水波方程的非振荡数值格式。该数值方法避免了模板的选取,计算量要比ENO方法小得多,而且仍具有高分辨率。第五章利用最小二乘思想,通过一次扩张模板来提高数值方法的精度,发展了在非结构网格上解二维浅水波方程的高精度有限体积法。构造数值算法过程中利用极值原理的思想,来保证解不出现新的极值,构造出解浅水波方程的三阶数值格式。该格式具有高分辨率,准确地模拟了间断现象。最后总结了四、五两章讨论的数值方法的优缺点,提出了今后工作的发展方向。更多还原

【Abstract】 The Shallow Water Equations arises in many applications such as oceanic, environment and bionomics. It is well known that the solutions of the Shallow Water Equations can be discontinuous even when the initial condition is smooth. This thesis is concerned with the least square finite volume method for Shallow Water Equations on unstructured meshes.The thesis consists of five chapters.In chapter 1, we briefly review the physical background, mathematical model and numerical methods for the Shallow Water Equations.In chapter 2, we present the numerical schemes for resolving Shallow Water Equations on structured and unstructured meshes, such as Godunov and composite schemes.In chapter 3, we present frame of the finite volume methods for 2-D hyperbolic conservation laws. Also we introduce control volume, TVD Runge-Kutta time discretization, maximum principle and limiters.In chapter 4, we present the non-oscillatory finite volume method for the Shallow Water Equations on unstructured meshes. We get the linear interpolation using the least square method and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the non-oscillatory numerical scheme. The method avoids choosing template, needs less computation and has high resolution of the discontinue.In chapter 5, we present the third-order finite volume method for the Shallow Water Equations on unstructured meshes. We get the secondary interpolation using the least square method by enlarging the template once and ensure that the solution will not produce the new extremum using the maximum principle. Finally we construct the third-order high resolution numerical scheme. It can capture discontinuous well.At last, we summarize the advantages and disadvantages in chapter4, 5, and show our task in the future.更多还原