Runge–Kutta Schemes for Addressing Left-Endpoint Singularities

In classical explicit Runge–Kutta methods for solving initial value problems (IVPs) of the form y′(x)=f(x,y),y(x0)=y0, the first stage is typically given by evaluating the right-hand side at the initial point, i.e., f1=f(x0,y0). However, this approach becomes inefficient or even ill-posed when the f(x0,y0) exhibits a singularity at x0, as is common in many physically motivated problems such as the Lane–Emden equation or Thomas–Fermi model. To address this issue, we propose an alternative approach that was originally introduced by Oliver for low-order methods. In this formulation, the first stage is shifted away from the singular point and is instead evaluated at a shifted location: f1=f(x0+c1τ,y0), where τ is the step size and c1≠0 is a nonzero coefficient. This allows the method to bypass the singularity while preserving consistency with the IVP. We derive the corresponding order conditions for algebraic order six and construct an eight-stage scheme satisfying these constraints. The resulting method demonstrates significantly improved efficiency when applied to problems with initial-point singularities, outperforming classical Runge–Kutta pairs of orders 6(5) and even 8(7).