Numerical Methods for Differential Systems

Numerical Methods for Differential Systems: Recent Developments in Algorithms, Software, and Applications reviews developments in algorithms, software, and applications of numerical methods for differential systems. Topics covered include numerical algorithms for ordinary and partial differential equations (ODE/PDEs); theoretical approaches to the solution of nonlinear algebraic and boundary value problems via associated differential systems; integration algorithms for initial-value ODEs with particular emphasis on stiff systems; finite difference algorithms; and general- and special-purpose computer codes for ODE/PDEs. Comprised of 15 chapters, this book begins with an introduction to high-order A-stable averaging algorithms for stiff differential systems, followed by a discussion on second derivative multistep formulas based on g-splines; numerical integration of linearized stiff ODEs; and numerical solution of large systems of stiff ODEs in a modular simulation framework. Subsequent chapters focus on numerical methods for mass action kinetics; a systematized collection of codes for solving two-point boundary value problems; general software for PDEs; and the choice of algorithms in automated method of lines solution of PDEs. The final chapter is devoted to quality software for ODEs. This monograph should be of interest to mathematicians, chemists, and chemical engineers.

Numerical Methods for Differential Systems: Recent Developments in Algorithms, Software, and Applications reviews developments in algorithms, software, and applications of numerical methods for differential systems. Topics covered include numerical algorithms for ordinary and partial differential equations (ODE/PDEs); theoretical approaches to the solution of nonlinear algebraic and boundary value problems via associated differential systems; integration algorithms for initial-value ODEs with particular emphasis on stiff systems; finite difference algorithms; and general- and special-purpose computer codes for ODE/PDEs. Comprised of 15 chapters, this book begins with an introduction to high-order A-stable averaging algorithms for stiff differential systems, followed by a discussion on second derivative multistep formulas based on g-splines; numerical integration of linearized stiff ODEs; and numerical solution of large systems of stiff ODEs in a modular simulation framework. Subsequent chapters focus on numerical methods for mass action kinetics; a systematized collection of codes for solving two-point boundary value problems; general software for PDEs; and the choice of algorithms in automated method of lines solution of PDEs. The final chapter is devoted to quality software for ODEs. This monograph should be of interest to mathematicians, chemists, and chemical engineers.

1;Front Cover;12;Numerical Methods for Differential Systems: Recent Developments in Algorithms, Software, and Applications;43;Copyright Page;54;Table of Contents;65;List of Contributors;86;Preface;107;Chapter 1. High-Order A-Stable Averaging Algorithmsfor Stiff Differential Systems;147.1;REFERENCES;368;Chapter 2. Second Derivative Multistep FormulasBased on g-Splines;388.1;1. INTRODUCTION;388.2;2. NOTATION AND DEFINITIONS;388.3;3. CONSTRUCTION OF FORMULAS;408.4;4. COMPARISONS OF FORMULAS;448.5;5. SUMMARY AND CONCLUSIONS;488.6;REFERENCES;509;Chapter 3. Numerical Integration of LinearizedStiff Ordinary Differential Equations;529.1;1. INTRODUCTION;529.2;2. THE NUMERICAL METHOD;529.3;3. APPLICATIONS;539.4;4. TEST RESULTS;559.5;5. CONCLUSIONS;579.6;REFERENCES;5710;Chapter 4. Comparing Numerical Methods for the Solution of Stiff Systems of ODEs Arisingin Chemistry;5810.1;1. INTRODUCTION;5810.2;2. TESTING NUMERICAL METHODS;6010.3;3. RESULTS AND RECOMMENDATIONS;6310.4;REFERENCES;7410.5;Appendix: Specification of Test Problems;7711;Chapter 5. On the Construction of Differential Systems for the Solution of Nonlinear Algebraic and Transcendental Systems ofEquations;8011.1;1. INTRODUCTION;8011.2;2. A GLOBAL ASYMPTOTIC STABILITY THEOREM;8211.3;3. DIFFERENTIAL METHODS LIMITED TO INVERTIBLE SYSTEMS;8511.4;4. DIFFERENTIAL METHODS FOR NON-INVERTIBLE SYSTEMS;8811.5;5. EXAMPLES AND CONVERGENCE RATES;9311.6;REFERENCES;9712;Chapter 6. Differential Procedures for Systems of Implicit Relations and Implicitly Coupled Nonlinear Boundary-ValueProblems;9812.1;1. INTRODUCTION AND PRELIMINARY CONSIDERATIONS;9812.2;2. GENERATION OF A SOLUTION ON A GRID IN;10012.3;3. GENERATION OF THE STARTING VALUES FOR THE GRID;10212.4;4. MULTIPLE BRANCH SOLUTIONS;10512.5;5. NONLINEAR BOUNDARY VALUE PROBLEMS WITH IMPLICIT COUPLING;10512.6;REFERENCES;10813;Chapter 7. Numerical Solution of Large Systems of Stiff Ordinary Differential Equationsin a Modular Simulation Framework;11013.1;1. INTRODUCTION;11013.2;2. NUMERICAL SOLUTION OF STIFF O.D.E.S.;11013.3;3. NUMERICAL SOLUTION OF SPARSE LINEAR ALGEBRAIC EQUATIONS;11113.4;4. NUMERICAL TESTING OF STIFF TECHNIQUES;11313.5;5. THE DYNSYS 2.0 EXECUTIVE PROGRAM;11813.6;6. EPILOGUE;11913.7;APPENDIX A: TEST EXAMPLES;12013.8;APPENDIX B: GEAR'S METHOD;12313.9;REFERENCES;12314;Chapter 8. FAST: A Translator for the Solution of Stiff and NonlinearDifferential and Algebraic Equations;13814.1;1. INTRODUCTION;13814.2;2. MATHEMATICAL CONDITIONS;13914.3;3. FAST TRANSLATOR;14214.4;4. THE NON-LINEARITIES;14714.5;5. DESCRIPTION OF THE TRANSLATOR'S OPERATION;14914.6;6. EXAMPLES;15014.7;CONCLUSIONS;15514.8;BIBLIOGRAPHY;15814.9;ACKNOWLEDGEMENTS;15915;Chapter 9. Applications of EPISODE: An Experimental Package for the Integration ofSystems of Ordinary Differential Equations;16015.1;1. Introduction;16015.2;2. Examples;16315.3;3. Package Description;17715.4;REFERENCES;17916;Chapter 10. SETKIN: A Chemical Kenetics Preprocessor Code;18016.1;1. INTRODUCTION;18016.2;2. EXAMPLE;18216.3;3. QKCALC(Qk (t) calculation);18316.4;4. BUSTER (Rate equation construction);18516.5;5. DIFFUN (Differential Equation Function);19016.6;6. PEDERV: (Jacobian Calculation);19116.7;7. SENSIT: (Sensitivity Analysis);19216.8;REFERENCES;19317;Chapter 11. Numerical Methods for Mass Action Kinetics;19417.1;1. INTRODUCTION;19417.2;2. SOME MATHEMATICAL PROPERTIES OF MASS ACTION KINETICS;19617.3;3. APPROXIMATE METHODS FOR MASS ACTION KINETICS;19917.4;4. CONCLUSIONS;20517.5;5. ACKNOWLEDGEMENTS;20617.6;REFERENCES;20618;Chapter 12. A Systematized Collection of Codes for SolvingTwo-Point Boundary-Value Problems;21018.1;1. Introduction;21018.2;2. Methods;21118.3;3. Integration Methods;21718.4;4. Codes;21918.5;5. Numerical Examples;22418.6;References;23819;Chapter 13. General Software for Partial Differential Equations;24219.1;1. Introduction;24219.2;2. Class of Problems;24319.3;3. Piecewis