A new algorithm for generating derangementsSpringer Science and Business Media LLC - - 1980
Selim G. Akl
A new algorithm for generating derangements based on a well known permutation generation method is presented and analysed. The algorithm is shown to be superior in storage and time requirements to the existing method.
A Projected Indefinite Dogleg Path Method for Equality Constrained OptimizationSpringer Science and Business Media LLC - - 1999
Jianzhong Zhang, Chengxian Xu
In this paper, we propose a 2-step trust region indefinite dogleg path method for the solution of nonlinear equality constrained optimization problems. The method is a globally convergent modification of the locally convergent Fontecilla method and an indefinite dogleg path method is proposed to get approximate solutions of quadratic programming subproblems even if the Hessian in the model is indefinite. The dogleg paths lie in the null space of the Jacobian matrix of the constraints. An ℓ1 exact penalty function is used in the method to determine if a trial point is accepted. The global convergence and the local two-step superlinear convergence rate are proved. Some numerical results are presented.
Semi-discrete Schwarz waveform relaxation algorithms for reaction diffusion equationsSpringer Science and Business Media LLC - Tập 54 - Trang 831-866 - 2014
Shu-Lin Wu, Mohammad D. Al-Khaleel
For time dependent problems, the Schwarz waveform relaxation (SWR) algorithm can be analyzed both at the continuous and semi-discrete level. For consistent space discretizations, one would naturally expect that the semi-discrete algorithm performs as predicted by the continuous analysis. We show in this paper for the reaction diffusion equation that this is not always the case. We consider two space discretization methods—the 2nd-order central finite difference method and the 4th-order compact finite difference method, and for each method we show that the semi-discrete SWR algorithm with Dirichlet transmission condition performs as predicted by the continuous analysis. However, for Robin transmission condition the semi-discrete SWR algorithm performs worse than predicted by the continuous analysis. For each type of transmission conditions, we show that the convergence factors of the semi-discrete SWR algorithm using the two space discretization methods are (almost) equal. Numerical results are presented to validate our conclusions.
Application of predictor-corrector schemes with several correctors in solving air pollution problemsSpringer Science and Business Media LLC - Tập 24 - Trang 700-715 - 1984
Zahari Zlatev
Systems of ordinary differential equations obtained by using splitting-up techniques in some air pollution models and a pseudospectral (Fourier) discretization of the first-order space derivatives are considered. The application of a fairly general class of predictor-corrector (PC) schemes in the time-discretization process is discussed. Several corrections with different corrector formulae are carried out in thesePC schemes. The classicalDahlquist theory valid for the case when the stepsize is constant is preserved (under very mild restrictions on the stepsize) when suchPC schemes are used as variable stepsize variable formula methods (VSVFM's). This fact is exploited by allowing the stepsize to follow the variation of a certain norm of the wind velocity vector in aVSVFM based on specially constructedPC schemes with large intervals of absolute stability on the imaginary axis. A device that attempts to check both the accuracy and the stability in the course of the integration process has been developed. The code based on the application of thisVSVFM in the time-integration part of the treatment of both 2-dimensional and 3-dimensional models has been tested by using meteorological data prepared at stations located in practically all European countries. The numerical results indicate thatPC schemes with several correctors can successfully be used for the class of problems under consideration. The main reason for this success is the special nature of the computational cost per time-step (due to the splitting approach used). Some short remarks on the possibility of extending the results for large systems ofODE's arising in the treatment of other classes of problems are made.
Implicit Euler and Lie splitting discretizations of nonlinear parabolic equations with delaySpringer Science and Business Media LLC - Tập 54 - Trang 673-689 - 2014
Eskil Hansen, Tony Stillfjord
A convergence analysis is presented for the implicit Euler and Lie splitting schemes when applied to nonlinear parabolic equations with delay. More precisely, we consider a vector field which is the sum of an unbounded dissipative operator and a delay term, where both point delays and distributed delays fit into the framework. Such equations are frequently encountered, e.g. in population dynamics. The main theoretical result is that both schemes converge with an order (of at least)
$$q=1/2$$
, without any artificial regularity assumptions. We discuss implementation details for the methods, and the convergence results are verified by numerical experiments demonstrating both the correct order, as well as the efficiency gain of Lie splitting as compared to the implicit Euler scheme.
The extrapolated first order method for solving systems with complex eigenvaluesSpringer Science and Business Media LLC - Tập 24 - Trang 357-365 - 1984
N. M. Missirlis
An extrapolated form of the basic first order stationary iterative method for solving linear systems when the associated iteration matrix possesses complex eigenvalues, is investigated. Sufficient (and necessary) conditions are given such that convergence is assured. An analytic determination of good (and sometimes optimum) values of the involved real parameter is presented in terms of certain bounds on the eigenvalues of the iteration matrix. The usefulness of the developed theory is shown through a simple application to the conventional Jacobi method.
Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations packageSpringer Science and Business Media LLC - - 2023
Gerd Steinebach
AbstractRosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family.