Simulation of nonlinear systems subject to modulated chirp signals

Purpose The purpose of the paper is to apply a novel technique for the simulation of nonlinear systems subject to modulated chirp signals. Design/methodology/approach The simulation technique is first described and its salient features are presented. Two examples are given to confirm the merits of the method. Findings The results indicate that the method is appropriate for simulating nonlinear systems subject to modulated chirp signals. In particular, the efficiency and accuracy of the method is seen to improve as the chirp frequency increases. In addition, error bounds are given for the method. Originality/value Chirp signals are employed in several important applications such as representing biological signals and in spread spectrum communications. Analysis of systems involving such signals requires accurate, appropriate and effective simulation techniques

On the efficiency of numerical homogenization methods

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. This was complemented by the question as to how structure preservation affects the long-time behaviour of numerical methods

Structure preserving Stochastic Impulse Methods for stiff Langevin systems with a uniform global error of order 1 or 1/2 on position

Impulse methods are generalized to a family of integrators for Langevin systems with quadratic stiff potentials and arbitrary soft potentials. Uniform error bounds (independent from stiff parameters) are obtained on integrated positions allowing for coarse integration steps. The resulting integrators are explicit and structure preserving (quasi-symplectic for Langevin systems)

A stroboscopic averaging algorithm for highly oscillatory delay problems

We propose and analyze a heterogenous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method (SAM) suggested here may provide approximations with \(\mathcal{O}(H^2+1/\Omega^2)\) errors with a computational effort that grows like \(H^{-1}\) (the inverse of the stepsize), uniformly in the forcing frequency Omega

An asymptotic parallel-in-time method for highly oscillatory PDEs

© 2014, Society for Industrial and Applied Mathematics. Available online at http://epubs.siam.org/doi/abs/10.1137/130914577We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable

From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials

We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way