Implicit, time-dependent variable grid finite difference methods for the approximation of a linear waterflood

Authors:
Jim Douglas and Mary Fanett Wheeler

Journal:
Math. Comp. **40** (1983), 107-121

MSC:
Primary 65M10; Secondary 35L65

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679436-8

MathSciNet review:
679436

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An implicit, time-dependent variable grid finite difference method based on the addition of an artificial diffusivity is introduced and analyzed for approximating the solution of a scalar conservation law in a single space variable. No relation between the grids at successive time steps is required for convergence. Two adaptive grid selection procedures are shown to be covered by the analysis. Analogous results are also established for an implicit upwinding procedure.

**[1]**J. Douglas, Jr., "Simulation of a linear waterflood,"*Free Boundary Problems*, vol. II, Istituto Nazionale di Alta Matematica "Francesco Severi", Roma, 1980. MR**630748 (83e:76078)****[2]**J. Douglas, Jr., B. L. Darlow, M. F. Wheeler & R. P. Kendall, "Self-adaptive finite element and finite difference methods for one-dimensional, two-phase, immiscible flow,"*SIAM J. Sci. Statist. Comput.*(To appear.)**[3]**S. N. Kružkov, "First order quasilinear equations in several indpendent variables,"*Math. USSR-Sb.*, v. 10, 1970, pp. 217-243.**[4]**P. D. Lax, "Shock waves and entropy,"*Contributions to Nonlinear Functional Analysis*(E. H. Zarantonello, ed.), Academic Press, New York, 1971. MR**0367471 (51:3713)****[5]**A. Y. Le Roux, "A numerical conception of entropy for quasi-linear equations,"*Math. Comp.*, v. 31, 1977, pp. 848-872. MR**0478651 (57:18128)****[6]**M. F. Wheeler,*A Self-Adaptive Finite Difference Procedure for One-Dimensional, Two-Phase, Immiscible Flow*, Seminar on Numerical Analysis and its Application to Continuum Physics, Coleção Atas, vol. 12, Rio de Janeiro, 1980. MR**590282 (82a:76084)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65M10,
35L65

Retrieve articles in all journals with MSC: 65M10, 35L65

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0679436-8

Article copyright:
© Copyright 1983
American Mathematical Society