Numerical Methods for Time-Dependent, Nonlinear Boundary Value Problems
Authors: Culham W.E. , Varga Richard S. 
Published: December 2007

Abstract
This paper presents and examines in detail extensions to the Galerkin method oil solution that make it numerically superior to conventional methods used to solve a certain class of time-dependent, nonlinear boundary value problems. This class of problems includes the equation that describes the flow of a fully compressible fluid in a porous medium. The Galerkin method with several different piecewise polynomial subspaces and a non-Galerkin piecewise polynomial subspaces and a non-Galerkin method specifically employing cubic spline functions are used to approximate the solution of a nonlinear parabolic equation with one spatial variable. With a parabolic equation with one spatial variable. With a known analytic solution of the problem, the accuracies of these approximations are determined and compared with conventional finite-difference approximations. Specially, the various methods are compared on the basis of the amount of computer time necessary to achieve a given accuracy, as well as with respect to the order oil convergence and computer core storage required. These tests indicate that the higher-order Galerkin methods require the least amount of computer time for a given range of accuracy.

Introduction
The purpose of this paper is to outline in detail the application of the Galerkin method, employing piecewise polynomials, to solve nonlinear piecewise polynomials, to solve nonlinear boundary-value problems and compare the computational efficiency of the Galerkin method with more conventional numerical methods. Numerical methods compared with the Galerkin technique include a non-Galerkin method that utilizes cubic spline interpolation and the conventional finite-difference methods. Four conventional time approximations were also studied in conjunction with the above mentioned space discretization methods. In an earlier paper, Price and Varga showed theoretically that higher-order approximations to certain semilinear convection-diffusion equations were possible by means of Galerkin techniques, but complete numerical results for such approximations were not given. Also, in a paper that introduced the Galerkin method to the petroleum industry, Price et al. demonstrated that higher-order approximations were far superior numerically to the conventional methods used to solve certain linear convection-diffusion type problems. Jennings, Douglas and Dupont and Douglas et al. have considered the application of Galerkin methods to various nonlinear problems, but again complete numerical results, problems, but again complete numerical results, including comprehensive comparisons with existing numerical methods, were not given. Thus, in addition to presenting some new and computationally efficient Galerkin formulations for nonlinear problems and numerically demonstrating their problems and numerically demonstrating their higher order accuracies, it was also desirable to test these. methods to determine if they also exhibited the same superiority in regard to computational efficiency as was demonstrated for the Galerkin methods applied to linear problems. If so, then the Galerkin technique could prove to be an important advancement toward developing faster numerical models for field application. To test and compare each method of solution, a problem involving the nonlinear gas-flow equation problem involving the nonlinear gas-flow equation in one spatial variable with a specific volumetric source term was chosen, for which a closed-form or analytic solution was known. Using this particular problem and its analytic solution, it was possible problem and its analytic solution, it was possible to determine numerically the order of convergence of each method, to compare each method on the basis of computer time expended to obtain a given accuracy, and to compare each method with respect to computer core storage required. In addition, the experimental data were used to define "consistent quadrature" and "consistent interpolation" schemes for the Galerkin methods. Finally, it was possible to formulate conclusions regarding the computational efficiency of the four time approximations investigated.
SPEJ
P. 374