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 one to use step sizes which are independent of the stiff part of the equation and also very efficient to implement. The methods are constructed by combining the leapfrog scheme with an explicit scheme with a larger stability region, an implicit scheme, or a trigonometric integrator. For spatially discretized partial differential equations, this class comprises local time-stepping schemes [J. Diaz and M. J. Grote, SIAM J. Sci. Comput., 31 (2009), pp. 1985-2014], [M. Grote, S. Michel, and S. Sauter, Math. Comp., 90 (2021), pp. 2603-2643] but also locally implicit or locally trigonometric integrators. Our main contribution is a rigorous error and stability analysis for the whole class of methods. Special emphasis is taken for explicit multirate methods, which are based on stabilized leapfrog-Chebyshev polynomials introduced in [C. Carle, M. Hochbruck, and A. Sturm, SIAM J. Numer. Anal., 58 (2020), pp. 2404-2433] as well as implicit/explicit (IMEX) methods based on theta-schemes.