Numerical oscillations on nonuniform grids (Q1370085)

The authors study a class of numerical methods to solve two-point boundary value problems on nonuniform grids. When standard centered finite difference formulas are generalized to nonuniform grids, the order of the truncation error is, generally, lower than on uniform grids. However, the use of some of these formulas may provide very accurate results. This surprising fact suggests that the global error should have an order of convergence greater than that of the truncation error. Such a phenomenon, which has been called supraconvergence, has received the attention of many authors. With the study of supraconvergence it becomes clear, that the truncation error does not provide a good indicator of the method's accuracy. Numerical experiments show that if one studies two finite difference discretizations for two-point boundary problems even if both are supraconvergent with the same global error order they may produce very different numerical simulations. The conclusion is, that the truncation and global error orders do not give enough information on the ``quality'' of the numerical simulation. If two formulas have the same global error order an indicator to distinguish them could be the size of the error constant. The boundedness properties of this constant are related to stability, but a more detailed analysis of its behavior can give important informations on the expected accuracy. The authors approach furnishes a prediction of the magnitude of numerical oscillations and also a study of the sensitivity of the method to the index of the node where a step change occurs.