The Numerical Solution of Parabolic and Elliptic Differential Equations

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Previous article Next article The Numerical Solution of Parabolic and Elliptic Differential EquationsD. W. Peaceman and H. H. Rachford, Jr.D. W. Peaceman and H. H. Rachford, Jr.https://doi.org/10.1137/0103003PDFPDF PLUSBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] G. H., Bruce, , D. W., Peaceman, , H. H., Rachford and , J. D. Rice, Calculation of unsteady-state gas flow through porous media, Trans. Amer. Inst. Mining and Met. Engrs., 198 (1953), 79– ISIGoogle Scholar[2] H. S. Carslaw and , J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947viii+386 MR0022294 Google Scholar[3] Stanley P. Frankel, Convergence rates of iterative treatments of partial differential equations, Math. Tables and Other Aids to Computation, 4 (1950), 65–75 MR0046149 CrossrefGoogle Scholar[4] George G. O'Brien, , Morton A. Hyman and , Sidney Kaplan, A study of the numerical solution of partial differential equations, J. Math. Physics, 29 (1951), 223–251 MR0040805 0042.13204 CrossrefISIGoogle Scholar[5] Jim Douglas, Jr., On the numerical integration of $\partial\sp 2u/\partial x\sp 2+\partial\sp 2u/\partial y\sp 2=\partial u/\partial t$ by implicit methods, J. Soc. Indust. Appl. Math., 3 (1955), 42–65 10.1137/0103004 MR0071875 0067.35802 LinkISIGoogle Scholar[6] Jim Douglas, Jr. and , T. M. Gallie, Jr., Variable time steps in the solution of the heat flow equation by a difference equation, Proc. Amer. Math. Soc., 6 (1955), 787–793 MR0078754 0066.10502 CrossrefGoogle Scholar[7] J. Douglas, Jr. and , D. W. Peaceman, Numerical solution of two-dimensional heat flow problems, to be presented at the May, 1955 meeting of The American Institute of Chemical Engineers at Houston, Texas Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A General Alternating-Direction Implicit Framework with Gaussian Process Regression Parameter Prediction for Large Sparse Linear SystemsKai Jiang, Xuehong Su, and Juan ZhangSIAM Journal on Scientific Computing, Vol. 44, No. 4 | 7 July 2022AbstractPDF (1383 KB)Convergence Analysis of the Nonoverlapping Robin--Robin Method for Nonlinear Elliptic EquationsEmil Engström and Eskil HansenSIAM Journal on Numerical Analysis, Vol. 60, No. 2 | 21 March 2022AbstractPDF (565 KB)Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDEM. RakhubaSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 4 March 2021AbstractPDF (935 KB)A Parallel Cyclic Reduction Algorithm for Pentadiagonal Systems with Application to a Convection-Dominated Heston PDEAbhijit Ghosh and Chittaranjan MishraSIAM Journal on Scientific Computing, Vol. 43, No. 2 | 29 April 2021AbstractPDF (643 KB)Operator Splitting for a Homogeneous Embedding of the Linear Complementarity ProblemBrendan O'DonoghueSIAM Journal on Optimization, Vol. 31, No. 3 | 4 August 2021AbstractPDF (695 KB)Adaptive Douglas--Rachford Splitting Algorithm from a Yosida Approximation StandpointZihan Liu and Kannan RamchandranSIAM Journal on Optimization, Vol. 31, No. 3 | 4 August 2021AbstractPDF (721 KB)Splitting with Near-Circulant Linear Systems: Applications to Total Variation CT and PETErnest K. Ryu, Seyoon Ko, and Joong-Ho WonSIAM Journal on Scientific Computing, Vol. 42, No. 1 | 4 February 2020AbstractPDF (1366 KB)Operator Splitting Performance Estimation: Tight Contraction Factors and Optimal Parameter SelectionErnest K. Ryu, Adrien B. Taylor, Carolina Bergeling, and Pontus GiselssonSIAM Journal on Optimization, Vol. 30, No. 3 | 13 August 2020AbstractPDF (569 KB)AMF-type W-methods for Parabolic Problems with Mixed DerivativesS. González-Pinto, E. Hairer, D. Hernández-Abreu, and S. Pérez-RodríguezSIAM Journal on Scientific Computing, Vol. 40, No. 5 | 13 September 2018AbstractPDF (702 KB)On MultiScale ADI Methods for Parabolic PDEs with a Discontinuous CoefficientZhilin Li, Xiaohong Chen, and Zhengru ZhangMultiscale Modeling & Simulation, Vol. 16, No. 4 | 18 October 2018AbstractPDF (7287 KB)On the Quasi-unconditional Stability of BDF-ADI Solvers for the Compressible Navier--Stokes Equations and Related Linear ProblemsOscar P. Bruno and Max CubillosSIAM Journal on Numerical Analysis, Vol. 55, No. 2 | 18 April 2017AbstractPDF (1278 KB)A Selective Linearization Method For Multiblock Convex OptimizationYu Du, Xiaodong Lin, and Andrzej RuszczyńskiSIAM Journal on Optimization, Vol. 27, No. 2 | 15 June 2017AbstractPDF (259 KB)Convergence Analysis of Douglas--Rachford Splitting Method for “Strongly + Weakly” Convex ProgrammingKe Guo, Deren Han, and Xiaoming YuanSIAM Journal on Numerical Analysis, Vol. 55, No. 4 | 6 July 2017AbstractPDF (1088 KB)Algebraic Multigrid Preconditioners for Multiphase Flow in Porous MediaQuan M. Bui, Howard C. Elman, and David J. MoultonSIAM Journal on Scientific Computing, Vol. 39, No. 5 | 26 October 2017AbstractPDF (511 KB)Fiber Orientation Distribution Estimation Using a Peaceman--Rachford Splitting MethodSIAM