Felix Kwok

Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media

We present a phase-based potential ordering that is an extension of the Cascade ordering introduced by Appleyard and Cheshire [John R. Appleyard, Ian M. Cheshire, The cascade method for accelerated convergence in implicit simulators, in: European Petroleum Conference, 1982, pp. 113-122]. The proposed ordering is valid for both two-phase and three-phase flow, and it can handle countercurrent flow due to gravity and/or capillarity. We show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully-implicit method (FIM) into one with only pressure dependence. The potential-based reduced Newton algorithm is then obtained by applying Newtons method to this reduced-order system. Numerical evidence shows that our potential-based reduced Newton solver is able to converge for time steps that are much larger than what the standard Newtons method can handle. In addition, whenever standard Newton converges, so does the reduced Newton algorithm, and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard Newtons method.

Scientific Computing - An Introduction using Maple and MATLAB

Scientific computing is the study of how to use computers effectively to solve problems that arise from the mathematical modeling of phenomena in science and engineering. It is based on mathematics, numerical and symbolic/algebraic computations and visualization. This book serves as an introduction to both the theory and practice of scientific computing, with each chapter presenting the basic algorithms that serve as the workhorses of many scientific codes; we explain both the theory behind these algorithms and how they must be implemented in order to work reliably in finite-precision arithmetic. The book includes many programs written in Matlab and Maple Maple is often used to derive numerical algorithms, whereas Matlab is used to implement them. The theory is developed in such a way that students can learn by themselves as they work through the text. Each chapter contains numerous examples and problems to help readers understand the material hands-on.

Least Squares Problems

Least squares problems appear very naturally when one would like to estimate values of parameters of a mathematical model from measured data which are subject to errors (see quote above). They appear however also in other con- texts, and form an important subclass of more general optimization problems, see Chapter 12. After several typical examples of least squares problems, we start in Section 6.2 with the linear least squares problem and the natural solution given by the normal equations.

Best Robin Parameters for Optimized Schwarz Methods at Cross Points

Optimized Schwarz methods are domain decomposition methods in which a large-scale PDE problem is solved by subdividing it into smaller subdomain problems, solving the subproblems in parallel, and iterating until one obtains a global solution that is consistent across subdomain boundaries. Fast convergence can be obtained if Robin conditions are used along subdomain boundaries, provided that the Robin parameters

Neumann–Neumann Waveform Relaxation for the Time-Dependent Heat Equation

p

Chladni Figures and the Tacoma Bridge: Motivating PDE Eigenvalue Problems via Vibrating Plates ∗

are chosen correctly. In the case of second-order elliptic problems such as the Poisson equation, it is well known for two-subdomain problems without overlap that the optimal choice is

Convergence of Implicit Monotone Schemes with Applications in Multiphase Flow in Porous Media

p = O(h^{-1/2})

Optimal Interface Conditions for an Arbitrary Decomposition into Subdomains

(where

On the Applicability of Lions’ Energy Estimates in the Analysis of Discrete Optimized Schwarz Methods with Cross Points

h

Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation for the Wave Equation

is the mesh size), with the resulting method having a convergence factor of