Error analysis of multirate leapfrog-type methods for second-order semilinear ODEs

In this paper we consider the numerical solution of second-order semilinear differential equations, for which the stiffness is induced by only a few components of the linear part. For such problems, the leapfrog scheme suffers from severe restrictions on the step size to ensure stability. We thus propose a general class of multirate leapfrog-type methods which allows to use step sizes which are independent on the stiff part of the equation and also very efficient to implement. This class comprises local time-stepping schemes [5, 7] but also locally implicit or locally trigonometric integrators. Our main contribution is a rigorous error and stability analysis with special emphasis on explicit multirate methods, which are based on stabilized leapfrog-Chebyshev polynomials introduced in [4].