Adaptive solution of BVPs in singularly perturbed second-order ODEs, by the extended Numerov method combined with an iterative local grid h-refinement

The extended Numerov scheme of Chawla is one of high-order compact finite-difference discretisations applicable to BVPs in second-order ODEs, and to PDEs. The use of such discretisations in connection with adaptive grids remains largely unexplored. In order to make up this deficiency, we study the combination of the extended Numerov discretisation with the iterative local adaptive grid h-refinement previously tested by the present author at the assumption of the conventional finite-difference discretisation. A detailed formalism enabling a posteriori error analysis of the extended Numerov scheme is developed, together with several alternative choices for error estimators, grid refinement indicators, and regridding strategies. The adaptive algorithms obtained are examined in calculations on a collection of 15 examples of singularly perturbed ODEs, including equations relevant for electrochemistry. The most satisfactory algorithm involves: deferred approach to error estimation, based on virtual grid coarsening for truncation error evaluation; local discretisation errors as refinement indicators; and regridding strategy that uses the mean indicator values as the indicator thresholds for refinement. The extended Numerov scheme proves more efficient than the conventional discretisation, when used together with the iterative h-refinement.