P-stable exponentially-fitted Obrechkoff methods of arbitrary order for second-order differential equations

We consider the construction of P-stable exponentially-fitted symmetric two-step Obrechkoff methods for solving second order differential equations related to an initial value problem. Our approach is based on two ideas: for the exponential fitting, we follow the ideas of Ixaru and Vanden Berghe; for the P-stability we introduce exponentially-fitted Padé approximants to the exponential function. By combining these two ideas, we are able to construct P-stable methods of arbitrary (even) order.