Generalized Collocation Methods for Time-Dependent, Nonlinear Boundary-Value Problems
Authors: Sincovec R.F. 
Published: December 2007

Abstract
This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow.

Introduction
The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods.
Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain.

PROBLEMS SOLVED PROBLEMS SOLVED
Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term.
SPEJ
P. 345