Numerical Methods for ODE IVPs

Numerical methods for ODE initial value problems are discussed: the forward and backward Euler methods, trapezoidal rule, predictor-corrector methods, Runge-Kutta methods, and TRBDF2, always analyzing consistency, stability, and convergence. Dynamic timestepping is introduced. The various numerical methods are applied to deterministic dynamical systems (harmonic oscillator, nonlinear pendulum, Van der Pol oscillator, Shaw oscillator, and Lorenz equations), including the development of chaotic solutions with strange attractors. For nonlinear ODEs, especially for stiff ODEs, TRBDF2 (plus Newton’s method) is recommended. Two proofs of the Equivalence Theorem for ODE initial value problems are given.