On Leapfrog-Chebyshev Schemes

Abstract

This paper is dedicated to the improvement of the efficiency of the leap-frog method for 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 that exhibit a much weaker CFL condition than the leap-frog method. However, these methods do not even approximately conserve the energy of the exact solution which can result in a bad approximation quality.In this paper we propose two remedies to this drawback. For linear problems we show by using energy techniques that damping the Chebyshev polynomial leads to approximations which approximately preserve a discrete energy norm over arbitrary long times. Moreover, with a completely different approach based on generating functions, we propose to use special starting values that considerably improve the stability. We show that the new schemes arising from these modifications are of order two and can be modified to be of order four. These convergence results apply to semilinear problems. Finally, we discuss the efficient implementation of the new schemes and give generalizations to fully nonlinear equations.

Stabilized leapfrog based local time-stepping method for the wave equation

“…We prove the equivalence of definition (2.22) and the algorithm in section 2.4 by using the following recursion for P ∆t p,ν , which is also noticed in [5].…”

“…Remark A.3. Let it be noted that some of the estimates of the following Lemmas A.4 and A.5 are also proven in [5]. More specifically, (A.8a) is proven in […”

“…f corresponds to the identity, i.e. a smaller time-step is applied everywhere, the above algorithm reduces to the leapfrog-Chebyshev formulation from [20, Corollary 3] for ν = 0, which was extended to the more general case ν ≥ 0 in [5].…”

“…We note that the idea of replacing standard Chebyshev polynomials by their stabilized version for added stability is well-known in the parabolic context [40,42]. It was also recently used to stabilize a multirate leapfrog-Chebyshev (discrete Gautschi-type) method for semilinear or nonlinear wave equations [5] and the Lagrange multiplier based LTS approach in [7].…”

A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients

“…We remark, however, since the minimum size of the mesh h min is smaller for a well resolved mesh, the computation is more expensive as the result of smaller time step due to the CFL condition. One possible remedy, which deserves further investigation, is the methods of local time stepping for which we refer to the recent works Carle et al [7], Grote et al [19] and the references therein.…”