【作者】 赵小红；

【导师】 李德生；

【作者基本信息】 燕山大学 ， 计算数学， 2008， 硕士

【摘要】 因为很多物理现象和过程的数学模型都可以用非线性偏微分方程来表示,而这些非线性偏微分方程在很多情况下,求解精确解比较困难,故非线性偏微分方程的数值解法在数值分析中占有重要的地位。近几十年来,它的理论和方法都有了很大的发展,而且在各个科学技术领域中的应用也越来越广泛。有限差分方法(FDM)作为一种数值离散方法,以其求解问题时的易操作性和较大的灵活性,在科学研究和工作计算中得到了广泛的应用。随着计算机技术的迅猛发展,如何精细准确地进行数值模拟已经成为重要课题。到目前为止,虽然出现了很多的有限差分方法,但是泰勒展开法作为一种比较经典,简单的方法,仍然在有限差分方程的数值求解中占有十分重要的地位。论文主要是应用泰勒展开法进行求解。全文共分为五章。第一章介绍了国内外有限差分方法的发展史及现状,并给出了论文研究的主要结果和创新点。第二章给出了论文所需要的一些预备知识。第三、四、五章分别给出了三个方程的有限差分格式,并且对每一种差分格式的收敛性和稳定性给出了证明,其中第四章、第五章给出了相应的数值例子,通过数值例子可以更加清楚地看到,所给出的格式的有效性和可行性。更多还原

【Abstract】 Because many mathematical models of the physical phenomenon and process all can be denoted by the nonlinear partial differential equations. In many causes, it is very difficult to solve the exact solution of these nonlinear PDES. Therefore, the numerical method of the PDES plays an important role in the numerical analysis. In these recent dozens of years, its theory and method have great development. Moreover, the application of its is more and more extensive in every technology field.As a numerical discreatization method, for its easy operation and great agility character in the solving problem, the finite difference has gotten extensive application in the science research and computation. With the rapid development of the computer, how to get the fine and true numerical simulation has become an important task. Untill now, there are a lot of finite difference methods, but Taylor expandness method is very important in the finite difference methods as a classical and simply method. This paper has proposed by Taylor expandness method.The full text altogether divides into five chapters. In first chapter, we introduce the finite difference theory history, the present situation and provide the typical study method and main result with innovation. The second chapter, some readiness knowledge is given. The third, fourth and fifth chapters, three finite difference schemes were proposed. Convergence and stability of the difference solution were proved. The fourth and fifth chapers, numerical results demonstrate that the methods proposing are efficient and reliable.更多还原