  In turbulent flows, local velocity differences often obey a cascade-like
hierarchical dynamics, in the sense that local velocity differences at a given
scale k are driven by deterministic and random forces from the next-higher
scale k-1. Here we consider such a hierarchically coupled model with periodic
boundary conditions, and show that it leads to an N-th order initial value
problem, where N is the number of cascade steps. We deal in detail with the
case N=7 and introduce a non-polynomial spline method that solves the problem
for arbitrary driving forces. Several examples of driving forces are
considered, and estimates of the numerical precision of our method are given.
We show how to optimize the numerical method to obtain a truncation error of
order O(h^5) rather than O(h^2), where h is the discretization step.
