Variable-step boundary value methods based on reverse Adams schemes and their grid redistribution (Q1902060)

The authors study the \(k\)-step reverse Adams boundary value methods (BVMs) on nonuniform grids and their grid redistribution for the initial value problem \(y'(t) = f(t,y)\), \(t \in T = [t_0, t_1]\), \(y(t_0) = y_0\), where \(y, y',f \in \mathbb{R}^m\) and \(f\) is as smooth as the analysis requires. The idea is to show that there exist high-order BVMs with good stability properties which work well with grid redistribution techniques. The paper discusses attainable convergence orders, conditioning of resulting discretization matrices and introduces a grid redistribution strategy based on equidistribution of the local truncation error. An algorithm for a solution of the initial value problem using \(k\)-step reverse Adams BVMs with grid redistribution is presented. Both linear and nonlinear problems with boundary layers are given to illustrate how much grid redistribution improves the reverse Adams BVMs.