【作者】 龙晓瀚；

【导师】 袁益让；

【作者基本信息】 山东大学 ， 计算数学， 2006， 博士

【摘要】 油藏的运移聚集、油气资源的开采、地下水的污染和海水入浸等众多的渗流问题，从物理本质上考虑，它们都是流体在地下复杂多孔介质问的运移现象。这类运移通常是对流占优或强对流占优的扩散(包括机械弥散)运动。这里我们具体考虑两个典型的渗流力学模型：可混容驱动问题和海水入浸问题。 对于上述大规模长时间的应用问题，数值方法求解其数学模型常常遇到很大的困难，显著的困难在于：强对流占优扩散方程的解具有陡峭且移动的前沿；求解一般抛物方程所采用的标准有限元或有限差分法会产生难以接受的数值震荡和数值弥散；为克服数值震荡，工业中求解这类问题所使用的加权迎风法又会产生较强的数值弥散，上世纪八十年代，美国著名计算数学家Jim Douglas等提出了修正特征方法(简称MMOC)，使用MMOC求解对流占优的扩散问题，很大程度上消除了传统方法导致的数值震荡和数值弥散，MMOC成功应用于诸如油藏模拟、海水入浸等问题。然而，特征方法也存在自身的不足，尤其是它不能保持物理问题所固有的质量守恒特性，在接下来的时间里，特征方法又得到了不断的改进，在原有的思想上提出了带修正平流项的特征方法(简称MMOCAA)，该方法能保持流体的质量守恒特性。然而，上述两种特征法均难以处理较复杂的边界条件，Euler-Lagrange局部共轭方法(简称ELLAM)，作为改进的特征方法不仅保持质量守恒，而且能够方便地处理边界条件。MMOCAA方法和ELLAM方法都已成功地应用于渗流问题的数值模拟。 作为提高运算效率的重要方法，多步方法广泛应用于常微分方程和抛物型偏微分方程的数值求解，用多步向后差分法(简称BDF)求解抛物问题，其离散程序是全隐格式，求解非线性抛物问题，隐格式所需的计算量较大，受稳定性限制，单纯的显格式又不实用，著名计算数学家Crouzeix于上世纪八十年代初提出了基于算子分解的隐-显多步有限元法，该法在最近得到了发展，成为求解某些抛物方程的高效方法，隐-显多步方法对时间的离散基于线性多步法，离散方程的一部分为隐格式，另一部分为显格式，其中要求隐格式为强A(O)-稳定。由Wildlund于上世纪六十年代提出的A(Θ)-稳定性，不仅克服了Dahlquist的A-稳定性的二阶限制，而且使得抛物方程的时间离散和空间离散可以独立进行。隐、显结合的多步方法求解抛物方程，每一个时间步所求解的线性方程组的系数矩阵是相同的，因而其格式非常高效，另外，这种方法还是稳定和相容的。 本文所做研究工作集中以下两个方面：首先，对两个典型的渗流问题提出并分析更多还原

【Abstract】 The transports of fluid in petroleum reservoir, seawater intrusion and many other physical phenomena in underground porous media are essentially convection-dominated flows. Numerical simulations of these problems involve solving nonlinear convection-dominated diffusion equations and usually encounter considerable practical difficulties. Convection-dominated diffusion equation possesses properties of hyperbolic one. Standard numerical methods solving these equations often produce nonphysical oscillations, and upwind methods usually cause excessive numerical diffusion which smear the sharp fronts. To seek numerical methods for such problems that reflect their almost hyperbolic nature, Douglas et al. de-velopcd the so-called modified method of characteristics (MMOC), which incorporate time stepping along the characteristics of the hyperbolic part of the parabolic equations.The MMOC procedure, however, fails to preserve integral conservation laws. Its variant, called the modified method of characteristics with adjusted advection (MMOCAA), preserves the conservation law at a minor additional computational cost. Both theoretical analysis and numerical results show that the MMOCAA has advantages over the MMOC; but both of them have difficulties in dealing with boundary condition. The Eulerian-Lagrangian localized adjoint method (ELLAM), as an eminent characteristic method, has two advantages over the MMOC : preserving the conservation law and treating general boundary conditions easily.Multistep methods have been extensively used as effective numerical methods for ODEs and parabolic PDEs. Backward difference methods, traditionally called backward difference formula (BDF), are basically implicit multistep methods. Implicit-explicit multistep methods, introduced by Crouzeix in the early 1980s, have been well developed to solve some kinds of parabolic problems in recent several years. These methods discretize time based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. Since their implementations require at each time step the solution of linear system with the same matrix for all time levels, the resulting schemes are very efficient. Furthermore, the schemes are stable and consistent.The major contributions of this dissertation are in the following two areas: First. we更多还原