Initial-value solvers

After the quick introduction to Euler's method that is part of the survey in chapter 1, devote three or four days of lecture to a complete development of fixed-step initial-value solvers:

• Day 1: Subsection 3.1.1, four interpretations of the Euler method: as path following through the direction field (illustrated by DELab (see section 5 below) or other interactive software), as the tangent to an exact solution curve, as an approximation to the derivative, as the first terms in a Taylor polynomial. Conclude with examples of accuracy, as in subsection 3.1.2.
• Day 2: Subsection 3.1.3, Better Methods: Use the geometric interpretation of Euler to motivate a better (second-order) method, Heun (RK2).

  Subsection 3.1.4, Global and Local Error: Use Taylor's theorem to analyze the local error in Euler and to demonstrate the superiority of Heun, one of several possible second-order Runge-Kutta methods.

  • []believe that the interpretation and analysis of Euler's method is one of the great opportunities to illustrate the real value of Taylor's theorem. Another is using Taylor as a tool for linearized analysis; e.g., example 48, section 6.8, p. 325. Taylor's theorem appears again in the discussion of finite difference approximations of derivatives in section 9.3, Boundary-Value Problems: Numerical Methods.
• Day 3: Subsection 3.1.5, An Even Better Method: Fourth-Order Runge-Kutta: Wave your hands to the effect that the Taylor series analysis that leads from Euler to RK2 can be continued (with a lot more algebra) to RK4, one version of which is stated on p. 99.

  Subsection 3.1.6, Numerical Stability, introduces a subtle but important idea. Certainly, numerical stability is one of the central pillars of a more mature understanding of initial-value solvers. An effective classroom approach is an interactive demonstration like that suggested in the MATLAB box on p. 101. An excursion into stiffness is irresistible at this point, but ....

To guide your assignment of exercises that support these lectures, note that the exercises in section 3.1 come in several varieties, including among others:

 
• Exercising a method and perhaps finding error, as in exercises 1-6 or 11-12
• One-step approximations that emphasize the nature of the underlying approximation, as in exercises 7-10.
• Exploring variations in error with step size, as in exercises 14-15, 25-26, 35-42, 45.
• Using numerical approximations to analyze behavior or data, as in exercises 16-23, 34.
• Analytic and geometric interpretations of the methods, as in exercises 27-32.
• Efficiency, as in exercises 13(b), 43-44.

For the sake of simplicity, all of the analysis in section 3.1 is presented for first-order scalar equations. Systems are regarded as a natural extension, as remarked on p. 103 of the text, and illustrated repeatedly, e.g., example 13, p. 18, in the quick introduction to Euler's method in chapter 1, Prologue.