A strategy to avoid ill-conditioned stars in the generalized finite difference method for solving one-dimensional problems

In this paper, we solve linear boundary value problems of second-order in ordinary differential equations with the generalized finite difference method and compare the numerical accuracy for different orders of approximations. We develop a strategy for dealing with ill-conditioned stars based on the condition number of the matrix of derivatives. In addition, we consider a scheme implemented with parallel processing for the formation of the stars and the calculation of the derivatives. We present some examples with high gradients in irregular discretizations exaggerated on purpose, to highlight the efficiency of the proposed strategy.