Chapter 9
Initial Value Problems for Ordinary Differential Equations
Initial value problems describe how a physical system evolves from a specified initial state. This chapter develops Euler and Runge–Kutta methods, error estimates, and adaptive step-size control, and examines stability and stiffness. It then extends these ideas to systems of first- and second-order differential equations and to time integration for particle dynamics, including the velocity Verlet method and its long-time behavior. Physical examples include cooling, oscillations, projectile motion with air resistance, and orbital motion.
In this chapter
- 9.1 Initial value problems in physics
- 9.2 Finite-difference approximation
- 9.3 Euler methods and basic time-stepping
- 9.4 Runge-Kutta methods
- 9.5 Error estimate and adaptive step-size control
- 9.6 Stability and stiffness
- 9.7 Beyond one-step: multistep methods (overview)
- 9.8 Systems of first-order differential equations
- 9.9 Reducing second-order equations to first-order systems
- 9.10 Euler method for second-order equations
- 9.11 Runge-Kutta methods for second-order equations
- 9.12 Step-size control for second-order ODEs
- 9.13 Systems of second-order differential equations
- 9.14 Time integration methods for particle dynamics
- 9.15 Physics applications: worked examples
- 9.16 Summary and practical guidelines
- 9.17 Computational exercises and projects
Additional notes
Supplementary explanations and derivations that extend selected topics from Chapter 9.
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Note 9.1 — Leapfrog and Velocity Verlet
Derivations from Taylor expansions, computational steps, and the relationship between the two methods.
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Note 9.2 — Why Verlet Methods Behave So Well
Time reversibility, symplectic integration, modified energy, and comparison with RK4.
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Note 9.3 — What Makes a Differential Equation Stiff?
Fast and slow time scales, stability restrictions of explicit methods, backward Euler, and Gear's backward differentiation methods.
Companion resources
Codes
Six programs in MATLAB, Python, C++, and Fortran: fixed-step Euler, Heun, and RK4; adaptive RKF45 for a single equation and for systems; damped spring motion; projectile motion; and fixed-step velocity Verlet for a gravitational orbit.
View Chapter 9 codes →Exercises & Projects
Additional exercises and physics projects for comparing time-stepping methods, testing error and stability, exploring physical models, and checking conserved quantities.
View Chapter 9 exercises & projects →Resources
Test equations with analytic solutions, method-selection guidance, numerical libraries, including solvers for stiff problems, and selected references.
View Chapter 9 resources →Updates & Errata
Corrections, clarifications, and post-publication updates associated with Chapter 9.
View Chapter 9 updates & errata →