A fifth order accurate geometric mesh finite difference method for general nonlinear two point boundary value problems

•A high order variable mesh method using finite difference approximations.•The variable mesh method shows superiority over uniform mesh method.•The order and accuracy of scheme has superiority over existing methods.•Applicable to the problems with significant first order derivatives.•Computationally fast algorithm and applicable to the boundary layer problems.

Second order boundary value problems are treated using fifth order accurate geometric mesh finite difference approximations. The method uses evaluation at two off steps, three neighbouring knots and it reduces to simple five terms recurrence relations. The formal convergence analysis and error estimate reveals that the truncation errors of the new discretizations are fifth order accurate. Some physical problems are solved to demonstrate efficiency and reliability of the proposed numerical method. The root mean square errors for different values of mesh ratio parameter have been given in tables to support the utility of the method.