Conditioning of IVPs¶

Consider the ODEs $u'=u$ and $u'=-u$ . In each case we compute $\partial f/\partial u = \pm 1$ , so the condition number bound is $e^{(b-a)}$ in both problems. However, they behave quite differently. In the case of exponential growth, $u'=u$ , the bound is the actual condition number.

using Plots

t = LinRange(0,3,800)
u = @. exp(t)*1
lower,upper = @. exp(t)*0.7,@. exp(t)*1.3
plot(t,u,l=:black,ribbon=(lower,upper),
    leg=:none,xlabel="t",ylabel="u(t)", title="Exponential divergence of solutions")

      
    

But with $u'=-u$ , solutions actually get closer together with time.

u = @. exp(-t)*1
lower,upper = @. exp(-t)*0.7,@. exp(-t)*1.3
plot(t,u,l=:black,ribbon=(lower,upper),
    leg=:none,xlabel="t",ylabel="u(t)", title="Exponential convergence of solutions")

In this case the actual condition number is one, due to the difference of solutions at the initial time.

Fundamentals of Numerical Computation

Conditioning of IVPs¶