The Numerical Solution of Parabolic and Elliptic Differential Equations

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 Journal on Imaging Sciences, Vol. 9, No. 2 | 3 May 2016AbstractPDF (1273 KB)Convergence Study on the Symmetric Version of ADMM with Larger Step SizesSIAM Journal on Imaging Sciences, Vol. 9, No. 3 | 22 September 2016AbstractPDF (836 KB)Isogeometric Preconditioners Based on Fast Solvers for the Sylvester EquationSIAM Journal on Scientific Computing, Vol. 38, No. 6 | 15 November 2016AbstractPDF (692 KB)Method of Lines Transpose: High Order L-Stable ${\mathcal O}(N)$ Schemes for Parabolic Equations Using Successive ConvolutionSIAM Journal on Numerical Analysis, Vol. 54, No. 3 | 2 June 2016AbstractPDF (1298 KB)Computational Methods for Linear Matrix EquationsSIAM Review, Vol. 58, No. 3 | 4 August 2016AbstractPDF (851 KB)A Proximal Strictly Contractive Peaceman--Rachford Splitting Method for Convex Programming with Applications to ImagingSIAM Journal on Imaging Sciences, Vol. 8, No. 2 | 24 June 2015AbstractPDF (1523 KB)On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear SystemsSIAM Journal on Numerical Analysis, Vol. 53, No. 4 | 14 July 2015AbstractPDF (364 KB)A Generalized Proximal Point Algorithm and Its Convergence RateSIAM Journal on Optimization, Vol. 24, No. 4 | 14 October 2014AbstractPDF (1948 KB)Fourth Order Accurate Scheme for the Space Fractional Diffusion EquationsSIAM Journal on Numerical Analysis, Vol. 52, No. 3 | 12 June 2014AbstractPDF (453 KB)A Strictly Contractive Peaceman--Rachford Splitting Method for Convex ProgrammingSIAM Journal on Optimization, Vol. 24, No. 3 | 17 July 2014AbstractPDF (930 KB)Balanced Splitting and Rebalanced SplittingSIAM Journal on Numerical Analysis, Vol. 51, No. 6 | 19 November 2013AbstractPDF (587 KB)A Convergence Analysis of the Peaceman--Rachford Scheme for Semilinear Evolution EquationsSIAM Journal on Numerical Analysis, Vol. 51, No. 4 | 2 July 2013AbstractPDF (226 KB)An $h$-Adaptive Operator Splitting Method for Two-Phase Flow in 3D Heterogeneous Porous MediaSIAM Journal on Scientific Computing, Vol. 35, No. 1 | 29 January 2013AbstractPDF (1869 KB)Domain Decomposition Approaches for Mesh Generation via the Equidistribution PrincipleSIAM Journal on Numerical Analysis, Vol. 50, No. 4 | 23 August 2012AbstractPDF (429 KB)Fast Multiple-Splitting Algorithms for Convex OptimizationSIAM Journal on Optimization, Vol. 22, No. 2 | 24 May 2012AbstractPDF (1115 KB)A Variational Approach for Sharpening High Dimensional ImagesSIAM Journal on Imaging Sciences, Vol. 5, No. 1 | 24 January 2012AbstractPDF (4775 KB)An LQP-Based Decomposition Method for Solving a Class of Variational InequalitiesSIAM Journal on Optimization, Vol. 21, No. 4 | 22 November 2011AbstractPDF (176 KB)An Error Analysis for Rational Galerkin Projection Applied to the Sylvester EquationSIAM Journal on Numerical Analysis, Vol. 49, No. 6 | 22 November 2011AbstractPDF (279 KB)Analysis of the Rational Krylov Subspace and ADI Methods for Solving the Lyapunov EquationSIAM Journal on Numerical Analysis, Vol. 49, No. 5 | 20 September 2011AbstractPDF (334 KB)Energy-Conserved Splitting Finite-Difference Time-Domain Methods for Maxwell's Equations in Three DimensionsSIAM Journal on Numerical Analysis, Vol. 48, No. 4 | 31 August 2010AbstractPDF (552 KB)Inverse Iteration for Purely Imaginary Eigenvalues with Application to the Detection of Hopf Bifurcations in Large-Scale ProblemsSIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 4 | 7 May 2010AbstractPDF (281 KB)From Functional Analysis to Iterative MethodsSIAM Review, Vol. 52, No. 2 | 6 May 2010AbstractPDF (279 KB)Nested Iterative Algorithms for Convex Constrained Image Recovery ProblemsSIAM Journal on Imaging Sciences, Vol. 2, No. 2 | 4 June 2009AbstractPDF (854 KB)General Projective Splitting Methods for Sums of Maximal Monotone OperatorsSIAM Journal on Control and Optimization, Vol. 48, No. 2 | 25 February 2009AbstractPDF (332 KB)Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and ConvergenceSIAM Journal on Optimization, Vol. 19, No. 3 | 31 October 2008AbstractPDF (335 KB)Efficient Preconditioning of Sequences of Nonsymmetric Linear SystemsSIAM Journal on Scientific Computing, Vol. 29, No. 5 | 28 September 2007AbstractPDF (242 KB)Robin–Robin Domain Decomposition Methods for the Stokes–Darcy CouplingSIAM Journal on Numerical Analysis, Vol. 45, No. 3 | 22 May 2007AbstractPDF (251 KB)An Alternating-Direction Implicit Orthogonal Spline Collocation Scheme for Nonlinear Parabolic Problems on Rectangular PolygonsSIAM Journal on Scientific Computing, Vol. 28, No. 3 | 4 August 2006AbstractPDF (241 KB)An Accelerated Splitting-up Method for Parabolic EquationsSIAM Journal on Mathematical Analysis, Vol. 37, No. 4 | 1 August 2006AbstractPDF (267 KB)An ADI-Like Preconditioner for Boltzmann TransportSIAM Journal on Scientific Computing, Vol. 26, No. 3 | 25 July 2006AbstractPDF (206 KB)Solving Degenerate Reaction-Diffusion Equations via Variable Step Peaceman--Rachford SplittingSIAM Journal on Scientific Computing, Vol. 25, No. 4 | 16 May 2012AbstractPDF (8559 KB)Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear SystemsSIAM Journal on Matrix Analysis and Applications, Vol. 24, No. 3 | 31 July 2006AbstractPDF (257 KB)Stabilized Explicit-Implicit Domain Decomposition Methods for the Numerical Solution of Parabolic EquationsSIAM Journal on Scientific Computing, Vol. 24, No. 1 | 25 July 2006AbstractPDF (216 KB)Difference Graphs of Block ADI MethodSIAM Journal on Numerical Analysis, Vol. 38, No. 3 | 26 July 2006AbstractPDF (162 KB)A Cyclic Low-Rank Smith Method for Large Sparse Lyapunov EquationsSIAM Journal on Scientific Computing, Vol. 21, No. 4 | 25 July 2006AbstractPDF (377 KB)Domain Decomposition Operator Splittings for the Solution of Parabolic EquationsSIAM Journal on Scientific Computing, Vol. 19, No. 3 | 25 July 2006AbstractPDF (406 KB)Discrete-time Orthogonal Spline Collocation Methods for Schrödinger Equations in Two Space VariablesSIAM Journal on Numerical Analysis, Vol. 35, No. 2 | 25 July 2006AbstractPDF (458 KB)An ADI Method for Hysteretic Reaction-Diffusion SystemsSIAM Journal on Numerical Analysis, Vol. 34, No. 3 | 25 July 2006AbstractPDF (459 KB)Application of ADI Iterative Methods to the Restoration of Noisy ImagesSIAM Journal on Matrix Analysis and Applications, Vol. 17, No. 1 | 17 February 2012AbstractPDF (2458 KB)Coordination in Coarse-Grained DecompositionSIAM Journal on Optimization, Vol. 4, No. 4 | 13 July 2006AbstractPDF (1888 KB)Alternating Direction Preconditioning for Nonsymmetric Systems of Linear EquationsSIAM Journal on Scientific Computing, Vol. 15, No. 2 | 13 July 2006AbstractPDF (1688 KB)Alternating Direction Implicit Iteration for Systems with Complex SpectraSIAM Journal on Numerical Analysis, Vol. 28, No. 3 | 14 July 2006AbstractPDF (1238 KB)Vectorization of the Odd–Even Hopscotch Scheme and the Alternating Direction Implicit Scheme for the Two-Dimensional Burgers EquationsSIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 2 | 13 July 2006AbstractPDF (1591 KB)The Solution of Two-Point Boundary Value Problems by the Alternating Group Explicit (AGE) MethodSIAM Journal on Scientific and Statistical Computing, Vol. 9, No. 3 | 13 July 2006AbstractPDF (639 KB)Tensor Product Generalized ADI Methods for Separable Elliptic ProblemsSIAM Journal on Numerical Analysis, Vol. 24, No. 1 | 14 July 2006AbstractPDF (1728 KB)Iterated Splitting Method of High Order for Time-Dependent Partial Differential EquationsSIAM Journal on Numerical Analysis, Vol. 21, No. 4 | 17 July 2006AbstractPDF (1855 KB)Multistep Splitting Methods of High Order for Initial Value ProblemsSIAM Journal on Numerical Analysis, Vol. 17, No. 3 | 17 July 2006AbstractPDF (1937 KB)Alternating Direction Implicit Methods for Parabolic Equations with a Mixed DerivativeSIAM Journal on Scientific and Statistical Computing, Vol. 1, No. 1 | 16 May 2012AbstractPDF (2857 KB)Splitting Algorithms for the Sum of Two Nonlinear OperatorsSIAM Journal on Numerical Analysis, Vol. 16, No. 6 | 17 July 2006AbstractPDF (1402 KB)Dynamic ADI Methods for Elliptic EquationsSIAM Journal on Numerical Analysis, Vol. 16, No. 5 | 17 July 2006AbstractPDF (2283 KB)The Alternating Phase Truncation Method for Numerical Solution of a Stefan ProblemSIAM Journal on Numerical Analysis, Vol. 16, No. 4 | 17 July 2006AbstractPDF (2605 KB)The Extrapolation of First Order Methods for Parabolic Partial Differential Equations. ISIAM Journal on Numerical Analysis, Vol. 15, No. 6 | 14 July 2006AbstractPDF (1229 KB)Numerical Solution of a Diffusion Consumption Problem with a Free BoundarySIAM Journal on Numerical Analysis, Vol. 12, No. 4 | 14 July 2006AbstractPDF (2304 KB)A Survey of Modern Numerical AnalysisSIAM Review, Vol. 15, No. 2 | 2 August 2006AbstractPDF (2464 KB)Iterative Solution of Implicit Approximations of Multidimensional Partial Differential EquationsSIAM Journal on Numerical Analysis, Vol. 5, No. 3 | 14 July 2006AbstractPDF (2718 KB)Rounding Errors in Alternating Direction Methods for Parabolic ProblemsSIAM Journal on Numerical Analysis, Vol. 5, No. 2 | 3 August 2006AbstractPDF (1267 KB)A New Computational Procedure for A.D.I. MethodsSIAM Journal on Numerical Analysis, Vol. 4, No. 2 | 14 July 2006AbstractPDF (593 KB)Multistage Alternating Direction MethodsSIAM Journal on Numerical Analysis, Vol. 3, No. 4 | 14 July 2006AbstractPDF (829 KB)Two-Level Difference Schemes for Hyperbolic SystemsSIAM Journal on Numerical Analysis, Vol. 3, No. 3 | 14 July 2006AbstractPDF (882 KB)A New Alternating Direction Method for Parabolic Equations in Three Space VariablesJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (641 KB)Nonsymmetric Difference EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 3 | 13 July 2006AbstractPDF (501 KB)Alternating Direction Schemes for the Heat Equation in a General DomainJournal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 3 | 14 July 2006AbstractPDF (1121 KB)The Solution of Elliptic Difference Equations by Semi-Explicit Iterative TechniquesJournal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 2, No. 1 | 3 August 2006AbstractPDF (1718 KB)An Alternating Direction Method for Operator EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 4 | 13 July 2006AbstractPDF (461 KB)The Numerical Solution of the Dirichlet Problem for Laplace’s Equation by Linear ProgrammingJournal of the Society for Industrial and Applied Mathematics, Vol. 12, No. 1 | 13 July 2006AbstractPDF (374 KB)Another Alternating-Direction-Implicit MethodJournal of the Society for Industrial and Applied Mathematics, Vol. 11, No. 4 | 13 July 2006AbstractPDF (290 KB)Numerical Studies of Transition From Laminar to Turbulent Flow Over a Flat PlateJournal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 4 | 13 July 2006AbstractPDF (2277 KB)On Incomplete Iteration for Implicit Parabolic Difference EquationsJournal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 3 | 10 July 2006AbstractPDF (466 KB)An Alternating-Direction-Implicit Iteration TechniqueJournal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 2 | 10 July 2006AbstractPDF (1508 KB)Simultaneous, Successive and Alternating Direction Iteration SchemesJournal of the Society for Industrial and Applied Mathematics, Vol. 8, No. 1 | 10 July 2006AbstractPDF (1569 KB)A Method of Block IterationJournal of the Society for Industrial and Applied Mathematics, Vol. 4, No. 4 | 10 July 2006AbstractPDF (667 KB)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.Journal of the Society for Industrial and Applied Mathematics, Vol. 3, No. 1 | 10 July 2006AbstractPDF (1365 KB) Volume 3, Issue 1| 1955Journal of the Society for Industrial and Applied Mathematics1-65 History Submitted:18 October 1954Published online:10 July 2006 InformationCopyright © 1955 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0103003Article page range:pp. 28-41ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics