Mathematical Research on Anomalous Diffusion Problem

反常扩散问题的数学研究

基本信息

批准号：
06650072
负责人：
MORI Masatake
金额：
$ 1.28万
依托单位：
School of Engineering, University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

项目摘要

Suppose that a uniform mixture of two metals is in an unstable nonequilibrium state and that the mixture evolves from the unstable state to a more stable nonuniform configuration consisting of two phases. Such a phenomenon is call an anomalous diffusion problem and time evolution of such a phenomenon is described by a nonlinear partial differential equation called the Cahn-Hilliard equation.The purpose of the present research project is to establish an efficient difference scheme for stable numerical solution of such kind of nonlinear partial differential equations. The idea of the present method is to start from discretization of the energy of the physical system under consideration and to apply a suitable discrete variation. The key point of the present research exists in this discrete variation. When we discretize the energy we defined the difference operators and the summation operators consistently with the differetial operators and the integration operator.Our method can be applied not only to the Cahn-Hilliard equation but also to a wide class of nonlinear partial differential equations, i. e. partial differential equations for energy dissipative system and for energy conservative system. Examples of equations for the dissipative system and the heat equation and the Cahn-Hilliard equation, and the wave equation and the KdV equation are examples of the equations for the conservative system. Our method were applied successfully to numerical solution of these equations. In the case of the Cahn-Hilliard equation we proved mathematically the stability of the scheme and the convergence of the numerical solution to the exact solution of the original equation.

假设两种金属的均匀混合物处于不稳定的非平衡状态，并且混合物从不稳定状态演变为由两相组成的更稳定的非均匀构型。这种现象称为反常扩散问题，这种现象的时间演化由称为 Cahn-Hilliard 方程的非线性偏微分方程描述。本研究项目的目的是建立一种有效的差分格式，用于稳定数值解此类非线性偏微分方程。本方法的思想是从所考虑的物理系统的能量的离散化开始，并应用合适的离散变化。本研究的重点就在于这种离散变化。当我们离散能量时，我们定义的差分算子和求和算子与微分算子和积分算子一致。我们的方法不仅可以应用于 Cahn-Hilliard 方程，还可以应用于一类广泛的非线性偏微分方程，i 。 e.能量耗散系统和能量保守系统的偏微分方程。耗散系统的方程示例以及热方程和Cahn-Hilliard方程以及波动方程和KdV方程是保守系统的方程示例。我们的方法成功地应用于这些方程的数值求解。对于 Cahn-Hilliard 方程，我们在数学上证明了该格式的稳定性以及数值解对原方程精确解的收敛性。