A nonstandard Euler scheme for y″+g(y)y′+f(y)y=0

We introduce a nonstandard Euler scheme for solving the differential equation y″+g(y)y′+f(y)y=0 which has the same linear stability properties as the differential equation and is conservative when g=0. The method is based on a physically motivated reduction of the equation to a system of two first-order equations and the use of Lie group integrators. The method is demonstrated on a few examples and compared to a standard MATLAB adaptive solver.