ABSTRACT

Dynamic systems may frequently be modeled by systems of ordinary differential equations (or ODEs) of the form

x˙(t) = f(x, t) x(t0) = x0 (5.1)

where x(t) ∈ Rn is a time-varying function that depends on the initial condition x0. Such problems are often referred to as “initial value problems.” A system of nonlinear differential equations cannot typically be solved analytically. In other words, a closed form expression for x(t) cannot be found directly from Equation (5.1), but rather must be solved numerically. In the numerical solution of Equation (5.1), a sequence of points x0, x1, x2 . . . ,

is computed that approximates the true solution at a set of time points t0, t1, t2, . . .. The time interval between adjacent time points is called the time step and an integration algorithm advances the numerical solution by one time step with each application. The time step hn+1 = tn+1 − tn may be constant for all time intervals over the entire integration interval t ∈ [t0, tN ] or may vary at each step. The basic integration algorithm advances the solution from tn to tn+1 with

integration step size hn+1 based on a calculation that involves previously computed values xn, xn−1, . . . and functions f (xn, tn) , f (xn−1, tn−1) , . . . . Each practical integration algorithm must satisfy certain criteria concerning

1. numerical accuracy,

2. numerical stability, and

3. numerical efficiency.