A finite difference approach for the numerical solution of non-smooth problems for Boundary Value ODEs

This paper concerns the numerical approximation of Boundary Value ODEs (BVPs) with non-smooth coefficients and solutions. Different strategies are presented to tackle the cases of known and unknown singularity locations. In the former, the original problem is transformed in a multipoint BVP and high order Extended Central Difference Formulas (ECDFs) are used to approximate the smooth branches of the solution and the Neumann boundary conditions (BCs) with same accuracy. In the latter, an iterative Hybrid method coupling ECDFs and the shooting technique has been introduced to approximate both the discontinuity point and the solution. Convergence analysis and numerical comparisons with other approaches from literature are also presented. Good performances in terms of errors and convergence order are reported by applying ECDFs and the Hybrid method to linear test BVPs and to a nonlinear bio-mechanical model, in both cases of mixed and Dirichlet BCs.