Monotone and oscillation matrices applied to finite difference approximations
Author:
Harvey S. Price
Journal:
Math. Comp. 22 (1968), 489-516
MSC:
Primary 65.65
DOI:
https://doi.org/10.1090/S0025-5718-1968-0232550-5
MathSciNet review:
0232550
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- Eduard Batschelet, Über die numerische Auflösung von Ranswertproblemen bei elliptischen partiellen Differentialgleichungen, Z. Angew. Math. Phys. 3 (1952), 165–193 (German). MR 60912, DOI https://doi.org/10.1007/bf02008824
- Garrett Birkhoff and Richard S. Varga, Implicit alternating direction methods, Trans. Amer. Math. Soc. 92 (1959), 13–24. MR 105814, DOI https://doi.org/10.1090/S0002-9947-1959-0105814-4
- Garrett Birkhoff, Richard S. Varga, and David Young, Alternating direction implicit methods, Advances in Computers, Vol. 3, Academic Press, New York, 1962, pp. 189–273. MR 0168131
- J. H. Bramble and B. E. Hubbard, On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of non-negative type, J. Math. and Phys. 43 (1964), 117–132. MR 162367
- James H. Bramble and Bert E. Hubbard, New monotone type approximations for elliptic problems, Math. Comp. 18 (1964), 349–367. MR 165702, DOI https://doi.org/10.1090/S0025-5718-1964-0165702-X L. Collatz, “Bermerkungen zur Fehlerabschätzung für das Differenzenverfahren bei partiellen Differentialgleichungen,” Z. Angew. Math. Mech., v. 13, 1933, pp. 56–57.
- Lothar Collatz, The numerical treatment of differential equations. 3d ed, Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. Translated from a supplemented version of the 2d German edition by P. G. Williams. MR 0109436
- David Feingold and Daniel Spohn, Un théorème simple sur les normes de matrices et ses conséquences, C. R. Acad. Sci. Paris 256 (1963), 2758–2760 (French). MR 150937
- George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124 F. P. Gantmacher & M. G. Krein, Oscillation Matrices and Small Vibrations of Mechanical Systems, GITTL, Moscow, 1950; English transl., Office of Technical Service, Dept. of Commerce, Washington, D. C. MR 14, 178. F. P. Gantmacher, The Theory of Matrices, Vol. II, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959. MR 16, 438; MR 21 #6372c. S. Gerschgorin, “Fehlerabschätzung für das Differenzenverfahren zur Lösung partieller Differentialgleichungen,” Z. Angew. Math. Mech., v. 10, 1930, pp. 373–382. S. Gerschgorin, “Über die Abrenzung der Eigenwerte einer Matrix,” Izv. Akad. Nauk SSSR Ser. Mat., v. 7, 1931, pp. 749–754.
- Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
- Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR 0175290
- Eugene Isaacson, Error estimates for parabolic equations, Comm. Pure Appl. Math. 14 (1961), 381–389. MR 137311, DOI https://doi.org/10.1002/cpa.3160140315
- Alexander Ostrowski, Determinanten mit überwiegender Hauptdiagonale und die absolute Konvergenz von linearen Iterationsprozessen, Comment. Math. Helv. 30 (1956), 175–210 (German). MR 76433, DOI https://doi.org/10.1007/BF02564340
- D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl. Math. 3 (1955), 28–41. MR 71874
- Carl Pearcy, On convergence of alternating direction procedures, Numer. Math. 4 (1962), 172–176. MR 145677, DOI https://doi.org/10.1007/BF01386310 H. S. Price, Monotone and Oscillation Matrices Applied to Finite Difference Approximations, Ph.D. Thesis, Case Institute of Technology, 1965. H. S. Price & R. S. Varga, Recent Numerical Experiments Comparing Successive Overrelaxation Iterative Methods with Implicit Alternating Direction Methods, Report No. 91, Gulf Research & Development Company, Reservoir Mechanics Division, 1962.
- Harvey S. Price, Richard S. Varga, and Joseph E. Warren, Application of oscillation matrices to diffusion-convection equations, J. Math. and Phys. 45 (1966), 301–311. MR 207230 M. L. Rockoff, Comparison of Some Iterative Methods for Solving Large Systems of Linear Equations, National Bureau of Standards Report No. 8577, 1964. W. H. Roudebush, Analysis of Discretization Error for Differential Equations with Discontinuous Coefficients, Ph.D. Thesis, Case Institute of Technology, 1963.
- E. L. Wachspress, Optimum alternating-direction-implicit iteration parameters for a model problem, J. Soc. Indust. Appl. Math. 10 (1962), 339–350. MR 150935
- Eugene L. Wachspress, Extended application of alternating direction implicit iteration model problem theory, J. Soc. Indust. Appl. Math. 11 (1963), 994–1016. MR 169373
- E. L. Wachspress and G. J. Habetler, An alternating-direction-implicit iteration technique, J. Soc. Indust. Appl. Math. 8 (1960), 403–424. MR 114308
- Olof B. Widlund, On the rate of convergence of an alternating direction implicit method in a noncommutative case, Math. Comp. 20 (1966), 500–515. MR 231551, DOI https://doi.org/10.1090/S0025-5718-1966-0231551-9
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502 D. M. Young & L. Ehrlich, “Some numerical studies of iterative methods for solving elliptic difference equations” in Boundary Problems in Differential Equations, Univ. of Wisconsin Press, Madison, Wis., 1960, pp. 143–162. MR 22 #5127.
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© Copyright 1968
American Mathematical Society