Alternating Direction Schemes Research Articles

Choice of Finite-Difference Schemes in Solving Coefficient Inverse Problems

Various choices of a finite-difference scheme for approximating the heat diffusion equation in solving a three-dimensional coefficient inverse problem were studied. A comparative analysis was conducted for several alternating direction schemes, such as locally one-dimensional, Douglas–Rachford, and Peaceman–Rachford schemes, as applied to nonlinear problems for the three-dimensional heat equation with temperature-dependent coefficients. Each numerical method was used to compute the temperature distribution inside a parallelepiped. The methods were compared in terms of the accuracy of the resulting solution and the computation time required for achieving the prescribed accuracy on a computer.

The Alternating-Direction Schemes and Numerical Analysis for the Three-dimensional Seawater Intrusion Simulation

Both numerical simulation and theoretical analysis of seawater intrusion in coastal regions are of great theoretical importance in environmental sciences. The mathematical model can be described as a coupled system of three dimensional nonlinear partial differential equations with initial-boundary value problems. In this paper, according to the actual conditions of molecular and three-dimensional characteristics of the problem, we construct the characteristic finite element alternating-direction schemes which can be divided into three continuous one-dimensional problems. By making use of tensor product algorithm, and priori estimation theory and techniques, the optimal order estimates in H 1 norm are derived for the error in the approximate solution.

Efficient orthogonal spline collocation methods for solving linear second order hyperbolic problems on rectangles

Piecewise Hermite bicubic orthogonal spline collocation Laplace-modified and alternating-direction schemes for the approximate solution of linear second order hyperbolic problems on rectangles are analyzed. The schemes are shown to be unconditionally stable and of optimal order accuracy in the \(H^1\) and discrete maximum norms for space and time, respectively. Implementations of the schemes are discussed and numerical results presented which demonstrate the accuracy and rate of convergence using various norms.

On Red-Black SOR Smoothing in Multigrid

Optimal relaxation parameters are obtained for red-black Gauss-Seidel relaxation in multigrid solvers of a family of elliptic equations. The resulting relaxation schemes are found to retain very high efficiency over an appreciable range of coefficients of the elliptic differential operator, yielding simple, inexpensive, and fully parallelizable smoothers in many situations where less cost-effective block- and alternating-direction schemes are commonly used.

Contractivity results for alternating direction schemes in Hilbert spaces

In this paper we obtain some contractivity results for operators R(− A< 1,…,− A n ), where R( z 1,…, z n ) are rational approximations to exp( z 1+⋯+ z n ), and A i are maximal monotone operators on a Hilbert space H. A general result is proved by using an extension to several variables of a result of Von Neumann for bounding f( A)( f a holomorphic function, A an operator on H). This theory is applied to the convergence analysis of Alternating Direction methods and, more generally, to Fractional Steps schemes.

Orthogonal Spline Collocation Laplace-Modified and Alternating-Direction Methods for Parabolic Problems on Rectangles

A complete stability and convergence analysis is given for two- and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplace-modified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.

Orthogonal spline collocation Laplace-modified and alternating-direction methods for parabolic problems on rectangles

A complete stability and convergence analysis is given for two- and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplace-modified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.

Mesh refinements for parabolic equations

Consider a finite difference approximation of the Crank-Nicolson type, using a local mesh refinement to an initial-boundary value problem for the parabolic equation m( x) u t = ( s( x) u x x + f( x, t). We consider the above difference approximation to u( x, t) on a grid pattern, where the middle third of the unit interval is refined with respect to the rest of the interval. Using the standard matrix arguments, we prove the matrix stability (spectral radius less than one) for the above class of problems. The local mesh refinement is achieved without loss of the second-order spatial accuracy, or of computational efficiency in solving the resulting equations. In two dimensions, we use our techniques to adapt the alternating direction schemes to a general mesh refinement pattern. Numerical experiments are presented which illustrate the computational effectiveness of the mesh refinement technique.

The Equivalence of Certain Alternating Direction and Locally One-Dimensional Difference Methods

Previous article Next article The Equivalence of Certain Alternating Direction and Locally One-Dimensional Difference MethodsA. R. Gourlay and Andrew R. MitchellA. R. Gourlay and Andrew R. Mitchellhttps://doi.org/10.1137/0706004PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] F. G. D'yakonov, On some difference schemes for solutions of boundary problems, U.S.S.R Comput. Math. and Math. Phys., 2 (1962), 55–77 10.1016/0041-5553(63)90270-2 0233.65052 CrossrefGoogle Scholar[2] G. Fairweather and , A. R. Mitchell, Some computational results of a new alternating direction procedure, Comput. J., 9 (1966), 298–303 CrossrefISIGoogle Scholar[3] G. Fairweather and , A. R. Mitchell, A new computational procedure for ${\rm A.D.I.}$ methods, SIAM J. Numer. Anal., 4 (1967), 163–170 10.1137/0704016 MR0218027 0252.65072 LinkGoogle Scholar[4] A. R. Gourlay and , A. R. Mitchell, A classification of split difference methods for hyperbolic equations in several space dimensions, SIAM J. Numer. Anal., 6 (1969), 62–71 10.1137/0706006 MR0260209 0175.16203 LinkISIGoogle Scholar[5] Bert E. Hubbard, Alternating direction schemes for the heat equation in a general domain, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 448–463 MR0196952 0138.09705 LinkGoogle Scholar[6] Bert Hubbard, Some locally one-dimensional difference schemes for parabolic equations in an arbitrary region, Math. Comp., 20 (1966), 53–59 MR0187415 0146.13801 CrossrefISIGoogle Scholar[7] R. B. Kellogg, Another alternating-direction-implicit method, J. Soc. Indust. Appl. Math., 11 (1963), 976–979 10.1137/0111071 MR0159414 0214.14703 LinkISIGoogle Scholar[8] Milton Lees, Alternating direction and semi-explicit difference methods for parabolic partial differential equations, Numer. Math., 3 (1961), 398–412 10.1007/BF01386038 MR0142212 0101.34101 CrossrefGoogle Scholar[9] A. A. Samarskii, On an economical difference method for the solution of a multidimensional parabolic equation in an arbitrary region, U.S.S.R. Comput. Math. and Math. Phys., 2 (1963), 894–926 10.1016/0041-5553(63)90504-4 0273.65078 CrossrefGoogle Scholar[10] A. A. Samarskii˘, Locally one-dimensional difference schemes on non-uniform grids, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 431–466 MR0223116 0273.65079 Google Scholar[11] A. A. Samarskii˘, An additivity principle for constructing efficient difference schemes, Dokl. Akad. Nauk SSSR, 165 (1965), 1253–1256 MR0207214 0158.15603 Google Scholar[12] R. Temam, Masters Thesis, Sur la stabilité et la convergence de la mèthode des pas fractionnaires, Doctoral thesis, University of Paris, Paris, 1967 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Analysis of a compact multi-step ADI method for linear parabolic equationInternational Journal of Modelling and Simulation, Vol. 40, No. 1 | 3 September 2018 Cross Ref Improved ADI Scheme for Linear Hyperbolic Equations: Extension to Nonlinear Cases and Compact ADI SchemesJournal of Scientific Computing, Vol. 72, No. 2 | 28 January 2017 Cross Ref Parabolic Partial Differential EquationsNumerical Solution of Partial Differential Equations in Science and Engineering | 25 February 2011 Cross Ref Elliptic Partial Differential EquationsNumerical Solution of Partial Differential Equations in Science and Engineering | 25 February 2011 Cross Ref Fundamental Schemes for Efficient Unconditionally Stable Implicit Finite-Difference Time-Domain MethodsIEEE Transactions on Antennas and Propagation, Vol. 56, No. 1 | 1 Jan 2008 Cross Ref A splitting up algorithm for the determination of the control parameter in multi dimensional parabolic problemApplied Mathematics and Computation, Vol. 166, No. 3 | 1 Jul 2005 Cross Ref A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problemNumerical Methods for Partial Differential Equations, Vol. 18, No. 5 | 21 August 2002 Cross Ref A Fractional Splitting Algorithm for Non-overlapping Domain DecompositionComputational Science — ICCS 2002 | 10 April 2002 Cross Ref Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problemsMathematics of Computation, Vol. 53, No. 187 | 1 January 1989 Cross Ref Splitting methods for fourth order parabolic partial differential equationsComputing, Vol. 37, No. 4 | 1 Dec 1986 Cross Ref An analysis of alternating-direction methods for parabolic equationsNumerical Methods for Partial Differential Equations, Vol. 1, No. 1 | 1 Jan 1985 Cross Ref Chapter 13 Collocation Solution of the Transport Equation Using a Locally Enhanced Alternating Direction FormulationUnification of Finite Element Methods | 1 Jan 1984 Cross Ref Time Discretization in Parabolic EquationsNumerical Solution of Partial Differential Equations: Theory, Tools and Case Studies | 1 Jan 1983 Cross Ref Numerical solution of the atmospheric diffusion equation for chemically reacting flowsJournal of Computational Physics, Vol. 45, No. 1 | 1 Jan 1982 Cross Ref Recent developments of the hopscotch ideaConference on the Numerical Solution of Differential Equations | 28 August 2006 Cross Ref The hopscotch class of difference methods for partial differential equationsNumerische Lösung nichtlinearer partieller Differential- und Integrodifferentialgleichungen | 24 August 2006 Cross Ref Splitting methods in partial differential equationsAbhandlungen aus dem Mathematischen Seminar der Universität Hamburg, Vol. 36, No. 1 | 26 November 2013 Cross Ref Volume 6, Issue 1| 1969SIAM Journal on Numerical Analysis1-159 History Submitted:06 February 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706004Article page range:pp. 37-46ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Some problems of difference scheme theory

1. DIFFERENCE schemes for solving various problems of mathematical physics have appeared in recent years. Economic methods for solving multidimensional problems have been particularly developed (alternating direction schemes, decomposition schemes, locally one-dimensional schemes etc.). A large number of different algorithms is now available for solving a given problem. Special interest is thus aroused by the theoretical and experimental comparison of difference schemes, the development of general principles for constructing families of difference schemes possessing given properties, and finding the schemes in the family that satisfy auxiliary optimality requirements (as regards accuracy, economy etc. 1. There are various schemes which give the same results (are algebraically identical) provided the right-hand sides and boundary conditions are matched in a certain way. In this case our choice of one scheme rather than another must be based on practical convenience. This situation is typical for economic methods of solving multidimensional problems. Several papers 11, 2, 3, 41 have been concerned with comparing economic methods. A comparison in [l, 21 with the operator decomposition method (for the equation of heat conduction in a rectangular region), to which the methods of [5 - 71 are reducible after eliminating the intermediate values, showed that the order of approximation of these methods depends, in the case of non-stationary boundary conditions (and for [71, in the case of stationary conditions also) on the method of specifying the boundary conditions at the intermediate step. It is shown that, by varying the right-hand side at the boundary base-points of the mesh, we can

Alternating Direction Schemes for the Heat Equation in a General Domain

Alternating Direction Schemes for the Heat Equation in a General Domain

Iterational alternating direction schemes for the numerical solution of the third boundary value problem in a p-dimensional parallelepiped

Iterational alternating direction schemes for the numerical solution of the third boundary value problem in a p-dimensional parallelepiped

The Numerical Solution of Parabolic and Elliptic Differential Equations

The Numerical Solution of Parabolic and Elliptic Differential Equations