# Exponential Integrator for Large Systems of (Stiff) ODEs

% Exponential Integrator for Large Systems of (Stiff) ODEs. % --------------------------------------------------------- % % This code is an exponential method of order 4 with variable step % size selection to solve equations of the form y' = f(y), t0 = y(t0). % A dense output formula to compute the solution at specified % grid points is also available. The computation of phi(A)*v = % ((exp(A)-I)/A)*v, where A is the Jacobian matrix of f and v is a % vector is done by Krylov techniques in the code EXP4 and exactly in % the code EXP4EX. Therefore EXP4 is the integrator for large systems % of ordinary differential equations (say, dimension > 100). % % For a quick start type 'help bruss2d', 'help laser', 'help heat2d' % or 'help rob1d' and follow the comments. % % The integrators EXP4 and EXP4EX have the same syntax as the MATLAB % ODE solvers. % % For a better description of the codes have a look at: % Marlis Hochbruck, Christian Lubich, Hubert Selhofer: "Exponential % Integrators for Large Systems of Differential Equations", % to appear in SIAM J. Sci. Comp. % % The latest versions of the code and the paper are available at % % http://na.uni-tuebingen.de/ % % FILES OF THE DISTRIBUTION: % -------------------------- % The two integrators: % * EXP4: The integrator for large problems with Krylov techniques. % * EXP4EX: The same integrator as EXP4 without Krylov techniques. % Utility files of the two integrators: % * ERRSTEP: Step size selection based on error estimates. % * EXP4_INITIALIZE: Script to initialize the integrators. % * H_INIT: Guess an initial step size. % * KRYSTEP: Step size selection based on the Krylov process. % * KRY_STEP_RED: Helper function for the Krylov processes. % * LANCZOS2: Compute the Lanczos approximation. % * ODEFUN: Wrapper to evaluate f and df/dy. % * PHIM: The function (exp(A)-I)/A for a matrix. % * PHI_ARN: The Arnoldi process. % * PHI_LAN: The Lanczos process for general A. % * PHI_LAN_SYM: The Lanczos process for symmetric A. % * PLOTBRUSS2D: Utility to visualize the solution of BRUSS2D. % * SC_NORM: Compute a scaled (weighted) norm. % * VPRINT: Conditionally print messages. % Info files: % * CHANGE.LOG.EXP4: The changes since the last release. % * CONTENTS: This file. % * COPYING: Please read this file. % Examples: % * BRUSS2D: Diffusion in 2D. % * HEAT2D: Heat equation in 2D. % * LASER: Model of an atom-laser-interaction. % * ROB1D: Chemical reaction in 1D. % % GETTING STARTED: % ---------------- % The best way is to look at the examples supplied. The easiest % example is the linear heat equation: HEAT2D. Use this file as a % prototype for your own problems (see also help odefile). ROB1D and % BRUSS2D are similar, but introduce a few more features. LASER is the % most advanced example: It uses an extension to the odefile standard % where no jacobian matrix has to be explicitly computed. This % function computes the product of the jacobian matrix times a vector % using FFTs without ever computing the jacobian itself. This may save % memory and computation time. % % The code EXP4EX is much easier to read and understand than the % Krylov based EXP4. % Be brave and follow the help for BRUSS2D and don't forget to look at % the movie. % % ATTENTION: % This code will NOT run under matlab 4.*. % Please read the file COPYING of the distribution carefully. % % Please send bug reports, experiences, wishes and comments to % exp4∂na.uni-tuebingen.de. % This file is part of the exp4 package available at % http://na.uni-tuebingen.de. % % This program is distributed in the hope that it will be useful, % but WITHOUT ANY WARRANTY; without even the implied warranty of % MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the % file COPYING of the distribution or send email to % exp4∂na.uni-tuebingen.de.

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