A hybrid asymptotic and augmented compact finite volume method for nonlinear singular two point boundary value problems

Abstract An accurate and efficient numerical method is proposed for nonlinear singular two point boundary value problems. The scheme combines Puiseux series asymptotic technique with augmented compact finite volume method. The main motivation is to get high order accurate solution for nonlinear singular problems. The key of the new method is to express the solution as the Puiseux series expansion on a small subinterval involving the singular point. The expansion contains an undetermined parameter which is an augmented variable in the numerical method. In this way, a nonlinear system is constructed in other subinterval and the parameter related with the semi-analytic solution near the singular point and the numerical solution can be simultaneously obtained. A rigorous error estimate for the solution of the singular differential equation is conducted and fourth order accuracy is obtained. Numerical examples confirm the theoretical analysis and efficiency of the new method.