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On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit Methods

1955; Society for Industrial and Applied Mathematics; Volume: 3; Issue: 1 Linguagem: Inglês

10.1137/0103004

2168-3484

Jim Douglas,

Numerical methods in inverse problems

Previous article On the Numerical Integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by Implicit MethodsJim Douglas, Jr.Jim Douglas, Jr.https://doi.org/10.1137/0103004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. H. Bruce, , D. W. Peaceman, , H. H. Rachford and , J. D. Rice, Calculations of unsteady-state gas flow through porous media, Journal of Petroleum technology, 198 (1953), 79–91 CrossrefGoogle Scholar[2] H. Lewy, , R. Courant and , K. Friedrichs, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100 (1928), 32–74 10.1007/BF01448839 MR1512478 CrossrefGoogle Scholar[3] J. Crank and , P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc., 43 (1947), 50–67 MR0019410 CrossrefISIGoogle Scholar[4] S. Hartree and , J. R. Womersley, A method for the numerical or mechanical solution of certain types of partial differential equations, Proc. Royal Soc. London, Series A, 161 (1937), 353–366 0017.08003 CrossrefGoogle Scholar[5] F. B. Hildebrand, On the convergence of numerical solutions of the heat-flow equation, J. Math. Physics, 31 (1952), 35–41 MR0048171 0048.07602 CrossrefISIGoogle Scholar[6] Fritz John, On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval, Comm. Pure Appl. Math., 5 (1952), 155–211 MR0047885 0047.33703 CrossrefISIGoogle Scholar[7] M. L. Juncosa and , D. M. Young, On the order of convergence of solutions of a difference equation to a solution of the diffusion equation, J. Soc. Indust. Appl. 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Ann., 102 (1930), 650–670 10.1007/BF01782368 MR1512599 CrossrefGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails Method of Lines Transpose: Energy Gradient Flows Using Direct Operator Inversion for Phase-Field ModelsMatthew Causley, Hana Cho, and Andrew Christlieb12 October 2017 | SIAM Journal on Scientific Computing, Vol. 39, No. 5AbstractPDF (2175 KB)Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive Convolution2 June 2016 | SIAM Journal on Numerical Analysis, Vol. 54, No. 3AbstractPDF (1298 KB)Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations12 June 2014 | SIAM Journal on Numerical Analysis, Vol. 52, No. 3AbstractPDF (453 KB)An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous Media29 January 2013 | SIAM Journal on Scientific Computing, Vol. 35, No. 1AbstractPDF (1869 KB)Difference Graphs of Block ADI Method26 July 2006 | SIAM Journal on Numerical Analysis, Vol. 38, No. 3AbstractPDF (162 KB)Domain Decomposition Operator Splittings for the Solution of Parabolic Equations25 July 2006 | SIAM Journal on Scientific Computing, Vol. 19, No. 3AbstractPDF (406 KB)Alternating Direction Implicit Methods for Parabolic Equations with a Mixed Derivative16 May 2012 | SIAM Journal on Scientific and Statistical Computing, Vol. 1, No. 1AbstractPDF (2857 KB)Iterative Solution of Implicit Approximations of Multidimensional Partial Differential Equations14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 3AbstractPDF (2718 KB)Rounding Errors in Alternating Direction Methods for Parabolic Problems3 August 2006 | SIAM Journal on Numerical Analysis, Vol. 5, No. 2AbstractPDF (1267 KB)Alternating Direction Schemes for the Heat Equation in a General Domain14 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 3AbstractPDF (1121 KB)On Incomplete Iteration for Implicit Parabolic Difference Equations10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 3AbstractPDF (466 KB)An Alternating-Direction-Implicit Iteration Technique10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2AbstractPDF (1508 KB)On the Relation Between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential Equations10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 4, No. 1AbstractPDF (1316 KB)The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1AbstractPDF (1199 KB) Volume 3, Issue 1| 1955Journal of the Society for Industrial and Applied Mathematics History Submitted:18 October 1954Published online:10 July 2006 InformationCopyright © 1955 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0103004Article page range:pp. 42-65ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics