Mmultistep Characteristic Methods For Numerical Simulations Of Porous Fluid Flow

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...