This paper is dedicated to the improvement of the efficiency of the leapfrog method for linear and semilinear second-order differential equations. In numerous situations the strict CFL condition of the leapfrog method is the main bottleneck that thwarts its performance. Based on Chebyshev polynomials new methods have been constructed for linear problems that exhibit a much weaker CFL condition than the leapfrog method (at a higher cost). However, these methods fail to produce the correct long-time behavior of the exact solution which can result in a bad approximation quality.
In this paper we introduce a new class of leapfrog-Chebyshev methods for semilinear problems. For the linear part, we use Chebyshev polynomials while the nonlinearity is treated by the standard leapfrog method. The method can be viewed as a multirate scheme because the nonlinearity is evaluated only once in each time step whereas the number of evaluations of the linear part corresponds to the degree of the Chebyshev polynomial. In contrast to existing literature (which is restricted to linear problems), we suggest to stabilize the scheme and we introduce a new starting value required for the two-step method.
A new representation formula for the approximations obtained by using generating functions allows us to fully understand the stability and the long-time behavior of the stabilized and the unstabilized scheme. In particular, for linear problems we prove that these new schemes approximately preserve a discrete energy norm over arbitrarily long times. The stability analysis shows that stabilization is essential to guarantee a favorable CFL condition for the multirate scheme, which is closely related to local time-stepping schemes. We also show error bounds of order two for semilinear problems and that a special choice of the stabilization yields order four for linear problems.
Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations.