Numerical methods for differential equations Book

Differential Equations Classification of Differential Equations Linear Equations Non-Linear Equations Existence and Uniqueness of Solutions Numerical Methods Computer Programming First Ideas and Single-Step Methods Analytical and Numerical Solutions A First Example The Taylor Series Method Runge-Kutta Methods Second and Higher Order Equations Error Considerations Definitions Local Truncation Error for the Taylor Series Method Local Truncation Error for the Runge-Kutta Method Local Truncation and Global Errors Local Error and LTE Runge-Kutta Methods Error Criteria A Third Order Formula Fourth Order Formulae Fifth and Higher Order Formulae Rationale for Higher Order Formulae Computational Examples Step-Size Control Steplength Prediction Error Estimation Local Extrapolation Error Estimation with RK Methods More Runge-Kutta Pairs Application of RK Embedding Dense Output Construction of Continuous Extensions Choice of Free Parameters Higher-Order Formulae Computational Aspects of Dense Output Inverse Interpolation Stability and Stiffness Absolute Stability Non-Linear Stability Stiffness Improving the Stability of RK Methods Multistep Methods The Linear Multistep Process Selection of Parameters A Third Order Implicit Formula A Third Order Explicit Formula Predictor-Corrector Schemes Error Estimation A Predictor-Corrector Program Multistep Formulae from Quadrature Quadrature Applied to Differential Equations The Adams-Bashforth Formulae The Adams-Moulton Formulae Other Multistep Formulae Varying the Step Size Numerical Results Stability of Multistep Methods Some Numerical Experiments Zero-Stability Weak Stability Theory Stability Properties of Some Formulae Stability of Predictor-Corrector Pairs Methods for Stiff Systems Differentiation Formulae Implementation of BDF Schemes A BDF Program Implicit Runge-Kutta Methods A Semi-Implicit RK Program Variable Coefficient Multistep Methods Variable Coefficient Integrators Practical Implementation Step-Size Estimation A Modified Approach An Application of STEP90
Global Error Estimation Classical Extrapolation Solving for the Correction An Example of Classical Extrapolation The Correction Technique Global Embedding A Global Embedding Program Second Order Equations Transformation of the RK Process A Direct Approach to the RKNG Processes The Special Second Order Problem Dense Output for RKN Methods Multistep Methods Partial Differential Equations Finite Differences Semi-Discretization of the Heat Equation Highly Stable Explicit Schemes Equations with Two Space Dimensions Non-Linear Equations Hyperbolic Equations Appendix A: Programs for Single Step Methods A Variable Step Taylor Method An Embedded Runge-Kutta Program A Sample RK Data File An Alternative Runge-Kutta Scheme Runge-Kutta with Dense Output A Sample Continuous RK Data File Appendix B: Multistep Programs A Constant Steplength Program A Variable Step Adams PC Scheme A Variable Coefficient Multistep Package Appendix C: Programs for Stiff Systems A BDF Program A Diagonally Implicit RK Program Appendix D: Global Embedding Programs The Gem Global Embedding Code The GEM90 Package with Global Embedding A Driver Program for GEM90
Appendix E: A Runge-Kutta Nyström Program Bibliography Index Each chapter also includes an introduction and a section of exercise problems.