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Multi-frequency, highly-oscillatory Hamiltonian problems derive from the mathematical modelling of many real life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems.

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Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi-discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.Keywords Multi-frequency highly-oscillatory problems · Stiffly-oscillatory problems · Hamiltonian problems · Energy-conserving methods · Spectral methods · Legendre polynomials · Hamiltonian Boundary Value Methods Mathematics Subject Classification (2010) 65P10 · 65L05 · 65N35 IntroductionMulti-frequency highly-oscillatory problems have been recently attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time [8]. The proposed approach has proven to be very efficient when solving a number of severe highly-oscillatory problems, allowing to effectively and accurately "resolve" all high-frequency components in the solution. Sometimes,

4In this paper we analyze the use of Chebyshev polynomials in distributed consensus applications. We 5 study the properties of these polynomials to propose a distributed algorithm that reaches the consensus 6 in a fast way. The algorithm is expressed in the form of a linear iteration and, at each step, the 7 agents only require to transmit their current state to their neighbors. The difference with respect to 8 previous approaches is that the update rule used by the network is based on the second order difference 9 equation that describes the Chebyshev polynomials of first kind. As a consequence, we show that our 10 algorithm achieves the consensus using far less iterations than other approaches. We characterize the 11 main properties of the algorithm for both, fixed and switching communication topologies. The main 12 contribution of the paper is the study of the properties of the Chebyshev polynomials in distributed 13 consensus applications, proposing an algorithm that increases the convergence rate with respect to 14 existing approaches. Theoretical results, as well as experiments with synthetic data, show the benefits 15 using our algorithm. 16