  Discrete integrable systems are closely related to orthogonal polynomials and
isospectral matrix transformations. In this paper, we use these relationships
to propose a nonautonomous time-discretization of the Camassa-Holm (CH) peakon
equation, which describes the motion of peakon waves, which are soliton waves
with sharp peaks. We then validate our time-discretization, and clarify its
asymptotic behavior as the discrete-time goes to infinity. We present numerical
examples to demonstrate that the proposed discrete equation captures peakon
wave motions.
