Scientific Computing with Ordinary Differential Equations

1 Time-Dependent Processes in Science and Engineering.- 1.1 Newton’s Celestial Mechanics.- 1.2 Classical Molecular Dynamics.- 1.3 Chemical Reaction Kinetics.- 1.4 Electrical Circuits.- Exercises.- 2 Existence and Uniqueness for Initial Value Problems.- 2.1 Global Existence and Uniqueness.- 2.2 Examples of Maximal Continuation.- 2.3 Structure of Nonunique Solutions.- 2.4 Weakly Singular Initial Value Problems.- 2.5 Singular Perturbation Problems.- 2.6 Quasilinear Differential-Algebraic Problems.- Exercises.- 3 Condition of Initial Value Problems.- 3.1 Sensitivity Under Perturbations.- 3.2 Stability of ODEs.- 3.3 Stability of Recursive Mappings.- Exercises.- 4 One-Step Methods for Nonstiff IVPs.- 4.1 Convergence Theory.- 4.2 Explicit Runge-Kutta Methods.- 4.3 Explicit Extrapolation Methods.- 5 Adaptive Control of One-Step Methods.- 5.1 Local Accuracy Control.- 5.2 Control-Theoretic Analysis.- 5.3 Error Estimation.- 5.4 Embedded Runge-Kutta Methods.- 5.5 Local Versus Achieved Accuracy.- Exercises.- 6 One-Step Methods for Stiff ODE and DAE IVPs.- 6.1 Inheritance of Asymptotic Stability.- 6.2 Implicit Runge-Kutta Methods.- 6.3 Collocation Methods.- 6.4 Linearly Implicit One-Step Methods.- Exercises.- 7 MultiStep Methods for ODE and DAE IVPs.- 7.1 Multistep Methods on Equidistant Meshes.- 7.2 Inheritance of Asymptotic Stability.- 7.3 Direct Construction of Efficient Multistep Methods.- 7.4 Adaptive Control of Order and Step Size.- Exercises.- 8 Boundary Value Problems for ODEs.- 8.1 Sensitivity for Two-Point EVPs.- 8.2 Initial Value Methods for Timelike EVPs.- 8.3 Cyclic Systems of Linear Equations.- 8.4 Global Discretization Methods for Spacelike EVPs.- 8.5 More General Types of BVPs.- 8.6 Variational Problems.- Exercises.- References.- Software.