Experiments with a local adaptive grid h-refinement for the finite-difference solution of BVPs in singularly perturbed second-order ODEs

Abstract

Despite many years of the development of adaptive grid methods, there is still a need for re-evaluation, tuning, and comparisons of the various approaches to grid adaptation. In this study we seek to identify the most reliable and computationally inexpensive variant of the iterative grid refinement/de-refinement algorithm for the finite-difference solution of singularly perturbed BVPs in second-order ODEs, that is also easily extendable to evolutionary PDEs. Conventional three-point discretisations of the first and second derivatives are used. Errors of the discrete solution and its first derivative are simultaneously controlled. The grid refinement is accomplished by adding nodes at inter-nodal positions, and de-refinement by removing previously added nodes in reverse order. Computational experiments are performed for 15 examples of ODEs, assuming several alternative choices of the a posteriori error estimators, grid refinement indicators, and regridding strategies. The most satisfactory results are obtained by combining: deferred approach to error estimation, based on higher-order reference derivative approximations for truncation error evaluation; interpolation-based indicator for grid refinement; and regridding strategy that uses the mean indicator value as the indicator threshold for refinement.