Numerical Solution of Ordinary Differential Equations

Abstract

Preface. Introduction. 1. Theory of differential equations: an introduction. 1.1 General solvability theory. 1.2 Stability of the initial value problem. 1.3 Direction fields. Problems. 2. Euler's method. 2.1 Euler's method. 2.2 Error analysis of Euler's method. 2.3 Asymptotic error analysis. 2.3.1 Richardson extrapolation. 2.4 Numerical stability. 2.4.1 Rounding error accumulation. Problems. 3. Systems of differential equations. 3.1 Higher order differential equations. 3.2 Numerical methods for systems. Problems. 4. The backward Euler method and the trapezoidal method. 4.1 The backward Euler method. 4.2 The trapezoidal method. Problems. 5. Taylor and Runge-Kutta methods. 5.1 Taylor methods. 5.2 Runge-Kutta methods. 5.3 Convergence, stability, and asymptotic error. 5.4 Runge-Kutta-Fehlberg methods. 5.5 Matlab codes. 5.6 Implicit Runge-Kutta methods. Problems. 6. Multistep methods. 6.1 Adams-Bashforth methods. 6.2 Adams-Moulton methods. 6.3 Computer codes. Problems. 7. General error analysis for multistep methods. 7.1 Truncation error. 7.2 Convergence. 7.3 A general error analysis. Problems. 8. Stiff differential equations. 8.1 The method of lines for a parabolic equation. 8.2 Backward differentiation formulas. 8.3 Stability regions for multistep methods. 8.4 Additional sources of difficulty. 8.5 Solving the finite difference method. 8.6 Computer codes. Problems. 9. Implicit RK methods for stiff differential equations. 9.1 Families of implicit Runge-Kutta methods. 9.2 Stability of Runge-Kutta methods. 9.3 Order reduction. 9.4 Runge-Kutta methods for stiff equations in practice. Problems. 10. Differential algebraic equations. 10.1 Initial conditions and drift. 10.2 DAEs as stiff differential equations. 10.3 Numerical issues: higher index problems. 10.4 Backward differentiation methods for DAEs. 10.5 Runge-Kutta methods for DAEs. 10.6 Index three problems from mechanics. 10.7 Higher index DAEs. Problems. 11. Two-point boundary value problems. 11.1 A finite difference method. 11.2 Nonlinear two-point boundary value problems. Problems. 12. Volterra integral equations. 12.1 Solvability theory. 12.2 Numerical methods. 12.3 Numerical methods - Theory. Problems. Appendix A. Taylor's theorem. Appendix B. Polynomial interpolation. Bibliography. Index.