【作者】 王素梅；

【导师】 姜子文；

【作者基本信息】 山东师范大学 ， 应用数学， 2008， 硕士

【摘要】 本文讨论了Sobolev方程在三角网格剖分下的扩展混合体积元方法和平面二维水沙模型的特征有限体积元方法，通过严格的数值分析，建立了相关的误差估计．第一章讨论Sobolev方程初边值问题：的扩展混合体积元方法。该方法综合了扩展方法可以直接应用到双压力方程、可以处理多个边值条件、适应于小参数微分方程、不需要求逆、可以解决微扩散和低渗透性微分方程的各种优点和混合体积元方法的高精度、计算简单、保持物理量间的局部守恒性等优点。本章针对三角形网格剖分，给出了扩展混合体积元的半离散格式和全离散格式的误差分析，得到了逼近未知函数、未知函数的梯度在L~2模及逼近流体的流量在L~2模和H(div)模下的最优误差估计。另外，利用正规Green函数法，还得到了关于未知函数及其导数的拟最优L~∞模误差估计。数值算例结果表明，我们得到的误差估计结果是合理有效的。第二章讨论平面二维水沙模型的特征有限体积元方法，该模型由如下方程组成水流连续方程水流运动方程泥沙连续方程河床变形方程这是一组以对流作用为主的水流泥沙运动方程，因此用特征有限体积元方法进行离散是恰当的。因为该方法综合了特征有限差分方法和特征有限元方法的主要优点，与特征有限差分方法相比，它保持了差分方法的计算简单性，而且具有网格剖分灵活、可以在不规则网格上计算的优点；与有限元方法相比，它具有计算量小、格式直观易于计算的优点，而且可以近似达到有限元的精度。更重要的一点是它保持了质量守恒，因此是一类很好的解决对流占优的水流泥沙方程的计算方法。然而，由于该模型的第一个方程具有强烈的双曲性质，因此在进行误差估计时，仅得到了真解与离散解的次优阶L~2模误差估计。更多还原

【Abstract】 In this paper, an expanded mixed covolume method on triangular grids for soboiev equation and a finite volume element method along characteristics for the 2-dimensional mathematic model of sediment transport equation are considered. By a thorough numerical analysis, the error estimates are obstained.In Chapter one. we consider the expanded mixed covolume method for the invitial-boundary value problem of sobolev equationThe method has the advantages both of expanded mixed element method and the mixed covolume element method. The method does not need the inverse, so it can be used to solve degenerate problems. Meanwhile, it satisfies the local conservation law with the discrete solution. By making the numerical analysis, we obtain the optimal error estimates of L~2 norm about the unknown function and its gradient as well as L~2 and H(div) norm about its flux. In addition, using of the regularized Green function, we obtain the quasi-optimal order estimates of L~∞norm about the unknown function and its derivative. The numerical example results show that the error estimates are effective.In Chapter two. we study a characteristic finite volume element method for the mathematic model consisting of following equations: Water continuity equationWater dynamic equationsSilt transport equationThe equation of bottom topography changeThe governing equations are uniformly convection-domained. so that it is appropriate to use the characteristic finite volume element method for discreting. Because the method not only has the advantages of the characteristic finite difference method but of the characteristic finite element. It has been widely used in comoutational fluid mechanics and heat transfer problems. It possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally. But the first equation of the model has the strongly hyperbolic nature, we only obstain the quasi-optimal L~2 norm error estimates between the exact solution and the discrete solution.更多还原