This is one of several papers by the authors and colleagues on this type of method. For the constant coefficient initial value problem \(y'=Ly+b(t)\), \(y(t_ 0)=y_ 0\) a boundary value method is used to approximate the solution on an interval \([t_ 0,T]\). By partitioning \([t_ 0,T]\) into \(k\) sub-intervals defined by \(t_ i=t_{i-1}+h_ i\), \(i=1,\ldots,k\), two-step approximations of the derivative over pairs of subintervals yield \(k-1\) equations, and a final one-step implicit formula gives an additional equation. This linear system can be solved to approximate the solution.
In this paper, the authors focus on stability when the steps are varying so that \(h_ i=rh_{i-1}\), \(r>1\). When all eigenvalues of \(L\) have negative real part, the continuous solution is bounded for \(t\geq t_ 0\). For such problems, it is desirable that the approximate solution is bounded as well. Using the well-conditioning of matrices defining the system, it is shown that the approximation obtained is bounded provided that the imaginary parts of eigenvalues of \(L\) are sufficiently small.
With the sequence of increasing steps selected for study, another aspect of this method is more critical. It may be shown for the last step that \(\lim_{k\to\infty} h_ k=(1-1/r)(T-t_ 0)\neq 0\). As this final step does not converge to zero (although \(h_ 1\) does), convergence of the proposed approximation to the continuous solution does not occur for many problems. Accordingly, this implementation of the boundary value method cannot be recommended as a general purpose method.