Numerical solution of second-order ordinary differential equations with singular source term using fourth-order Runge-Kutta methods

Abstract

This work numerically investigates initial value problems governed by ordinary differential equations with source terms of Dirac delta type, using a direct approach based on classical Runge--Kutta methods. In contrast to the literature on impulsive systems, where singularities are handled through jump conditions at known times, the singularity is explicitly treated via a regularization of the Dirac delta. Five approximation functions are considered: one with non-compact support and four with compact support, the latter being widely used in fluid--structure interaction problems within the framework of Peskin's immersed boundary method. The classical RK4 and the RK46 methods were implemented in the C programming language and first validated using test problems with smooth solutions, for which both methods achieved the expected fourth-order convergence. When applied to problems with singular sources, the results reveal a significant loss of convergence order, with observed values below one, indicating limitations of the direct approach. Nevertheless, the RK46 method showed slightly better performance than RK4 in terms of absolute error, and the regularization function with non-compact support yielded the best results among the approximations considered. These findings provide a systematic assessment of the behavior of classical methods in the presence of directly treated singularities and highlight the need for the development of more robust numerical schemes for this class of problems. They also motivate further investigation into alternative regularization functions for approximating the Dirac delta, aiming at improved numerical accuracy.