On numerical methods for highly oscillatory problems in circuit simulation

We propose in this paper a novel technique for an efficient numerical approximation of systems of highly oscillatory ordinary differential equations. In particular, we consider electronic systems subject to modulated signals. A Filon-type method is proposed for use and compared with traditional trapezoidal rule and Runge–Kutta methods. The Filontype method is combined with the waveform relaxation technique for nonlinear systems. Preliminary numerical examples highlight the efficacy of this approach

Numerical methods for systems of highly oscillatory ordinary differential equations

Current research made contribution to the numerical analysis of highly oscillatory ordinary differential equations. Highly oscillatory functions appear to be at the forefront of the research in numerical analysis. In this work we developed efficient numerical algorithms for solving highly oscillatory differential equations. The main important achievements are: to the contrary of classical methods, our numerical methods share the feature that asymptotically the approximation to the exact solution improves as the frequency of oscillation grows; also our methods are computationally feasible and as such do not require fine partition of the integration interval. In this work we show that our methods introduce better accuracy of approximation as compared with the state of the art solvers in Matlab and Maple.This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations. Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically, our methods depend on inverse powers of the frequency of oscillation, turning the major computational problem into an advantage. Evolving ideas from the stationary phase method, we first apply the asymptotic method to solve highly oscillatory linear systems of differential equations. The asymptotic method provides a background for our next, the Filon-type method, which is highly accurate and requires computation of moments. We also introduce two novel methods. The first method, we call it the FM method, is a combination of Magnus approach and the Filon-type method, to solve matrix exponential. The second method, we call it the WRF method, a combination of the Filon-type method and the waveform relaxation methods, for solving highly oscillatory non-linear systems. Finally, completing the theory, we show that the Filon-type method can be replaced by a less accurate but moment free Levin-type method.The work was supported by Trinity College, University of Cambridge

Efficient computation of delay differential equations with highly oscillatory terms.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation

A new simulation technique for RF oscillators

The study of phase-noise in oscillators and the design of new circuit topologies necessitates an efficient technique for the simulation of oscillators. While numerous approaches have been developed over the years e.g. [1-3], each has its own merits and demerits. In this contribution, an asymptotic numeric method developed in e.g. [4-5] is applied to the simulation of RF oscillators. The method is closely related to the stroboscopic and high-order averaging method in [6] and the Heterogeneous Multiscale Methods in [7]. The method is advantageous in that the same methodology can be applied for the simulation of general circuit problems involving highly oscillatory ordinary differential equations, partial differential equations and delay differential equations. Furthermore and counter-intuitively, its efficacy improves with increasing frequency, a feature that is very favourable in modern communications systems where operating frequencies are ever rising. Results for a CMOS oscillator will confirm the validity and efficiency of the proposed method

Numerical integrators for motion under a strong constraining force

This paper deals with the numerical integration of Hamiltonian systems in which a stiff anharmonic potential causes highly oscillatory solution behavior with solution-dependent frequencies. The impulse method, which uses micro- and macro-steps for the integration of fast and slow parts, respectively, does not work satisfactorily on such problems. Here it is shown that variants of the impulse method with suitable projection preserve the actions as adiabatic invariants and yield accurate approximations, with macro-stepsizes that are not restricted by the stiffness parameter