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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
MathSciNet review: 679436
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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.

References [Enhancements On Off] (What's this?)

  • [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)

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Article copyright: © Copyright 1983 American Mathematical Society

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