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

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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.

**[1]**Jim Douglas Jr.,*Simulation of a linear waterflood*, Free boundary problems, Vol. II (Pavia, 1979) Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980, pp. 195–216. MR**630748****[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]**Peter D. Lax,*The formation and decay of shock waves*, Visiting scholars’ lectures (Texas Tech Univ., Lubbock, Tex., 1970/71), Texas Tech Press, Texas Tech Univ., Lubbock, Tex., 1971, pp. 107–139. Math. Ser., No. 9. MR**0367471****[5]**A. Y. le Roux,*A numerical conception of entropy for quasi-linear equations*, Math. Comp.**31**(1977), no. 140, 848–872. MR**0478651**, https://doi.org/10.1090/S0025-5718-1977-0478651-3**[6]**Mary Fanett Wheeler,*A self-adaptive finite difference procedure for one-dimensional, two-phase, immiscible flow*, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 153–161. MR**590282**

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

Article copyright:
© Copyright 1983
American Mathematical Society