This module explores the stability of Euler's method for solving
initial value problems for ordinary differential equations. A
numerical method for an ordinary differential equation (ODE) generates
an approximate solution step-by-step in discrete increments across the
interval of integration, in effect producing a discrete sample of
approximate values of the solution function. Such a numerical method
is said to be stable if small perturbations do not cause the
resulting numerical solutions to diverge without bound. This module
applies Euler's method to solve two-dimensional homogeneous linear
systems of ODEs with constant coefficients. Such a system has the form
The user begins by clicking Random to generate a random† coefficient matrix and initial value. The generated coefficient matrix A is printed below, and the two components of the initial value are marked on the graph by red and blue dots. The components of the exact solution to the initial value problem are drawn on the graph with colors corresponding to those of the components of the initial value. Starting from the initial value, the user advances the numerical solution through successive steps using Euler's method. Each step of Euler's method is presented as a three-stage process. Each stage is executed by clicking either Next or the currently highlighted stage:
Successive steps can be continued until the the interval has been fully traversed. The user can click Reset to start over with the current initial value problem or Random to generate a new initial value problem.
†The randomly generated constant coefficient matrix is not random in the sense of randomly generated entries. Rather, the module strives to construct a matrix A with entries between −10 and 10 such that the eigenvalues of hk A fall inside the stability region for sufficiently small step sizes but outside it for permitted large step sizes.
Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 9.3.2.
Developers: Evan VanderZee and Michael Heath