1.1 Overview

1.1.1 Measurable outcomes

In this module we will introduce the numerical solution to an ordinary differential equation (ODE). While some differential equations, like many of those you saw in 18.03SC Differential Equations, have analytical solutions, there are many interesting ODEs that do not have analytical solutions. For those problems without analytical solutions we will use numerical methods to approximate the solution.

Specifically, students successfully completing this module will be able to:

  • Measurable Outcome 1.1: Define a first-order ODE.

  • Measurable Outcome 1.2: Use analytical solutions to validate numerical solutions of ODE.

  • Measurable Outcome 1.3: Distinguish nonlinear ODEs from linear ODEs.

  • Measurable Outcome 1.4: Approximate the behavior of a nonlinear equation with a linear one.

  • Measurable Outcome 1.5: Discretize a univariate function and its derivative, assess the truncation error using Taylor series analysis.

  • Measurable Outcome 1.6: Describe the Forward Euler methods and the Midpoint methods.

  • Measurable Outcome 1.7: Assess whether a numerical method converges, and calculate its global order of accuracy.

  • Measurable Outcome 1.8: Assess whether a numerical method is consistent, and calculate its local order of accuracy.

  • Measurable Outcome 1.9: Assess whether a numerical method is zero stable, and whether a numerical method is eigenvalue stable.

  • Measurable Outcome 1.10: Demonstrate an understanding of the Dahlquist Equivalence Theorem by describing the relationship between a convergent method, consistency, and stability.

  • Measurable Outcome 1.11: Explain the concept of stiffness of a system of equations.

  • Measurable Outcome 1.12: Describe how stiffness impacts the choice of numerical method for solving the equations.

  • Measurable Outcome 1.13: Explain the differences and relative advantages between explicit and implicit methods to integrate systems of ordinary differential equations.

  • Measurable Outcome 1.14: For nonlinear systems of equations, explain how a Newton-Raphson can be used in the solution of an implicit method.

  • Measurable Outcome 1.15: Describe multi-step methods, including the Adams-Bashforth, Adams-Moulton, and backwards differentiation families.

  • Measurable Outcome 1.16: Describe the form of the Runge-Kutta family of multi-stage methods.

  • Measurable Outcome 1.17: Explain the relative computational costs of multi-step versus multi-stage methods.

  • Measurable Outcome 1.18: Determine the stability boundary for a multi-step or multi-stage method applied to a linear system of ODEs.

  • Measurable Outcome 1.19: Recommend an appropriate ODE integration method based on the features of the problem being solved.