Dissipative Chebyshev exponential-fitted methods for numerical solution of second-order differential equations

A family of implicit methods based on intra-step Chebyshev interpolation has been developed to integrate oscillatory second-order initial value problems of the form y"(t)- 2g y'(t)+ (g2+ w2)y(t)= f(t,y(t)). The procedure integrates the homogeneous part exactly (in the absence of round-off errors). The Chebyshev approach uses stepsizes that are considerably larger than those typically used in Runge-Kutta or multistep methods. Computational overheads are comparable to those incurred by high-order conventional procedures. Chebyshev interpolation coupled with the exponential-fitted nature of the method substantially reduces local errors. Global error propagation rates are also reduced making these procedures good candidates to be used in long-term simulations of perturbed oscillatory systems with a dissipative term.