UC Berkeley

We develop methods for numerically integrating oscillatory functions and employ one of the methods in a new spectrally accurate collocation method to solve the linearized Vlasov-Poisson equation describing a collisionless plasma.

In 1923, Filon provided one of the first methods for evaluating definite oscillatory integrals via a quadrature formula. Filon's method involved treating the integral as a non-oscillatory function being multiplied by an oscillatory function. An inherent difficulty to what are now known as Filon methods involves integrating a set of basis functions against the oscillatory part of the function. We present two moment free Filon method capable of calculating indefinite integrals on a finite interval for irregular oscillators. These are the first moment free Filon methods with a polynomial basis that can compute integrals for general irregular oscillators. This work is a generalization of previous work by Hasegawa and Torii for the regular oscillator case that relies on representing the non-oscillatory part of the function via shifted Chebychev polynomials. We also present a novel method for calculating the regular oscillator case.

We then present a new collocation technique of computing solutions of the single-mode 1-D Vlasov-Poisson problem in which the plasma density is represented on Chebychev panels using high-order polynomials in time. We apply this method to a Volterra equation first discovered by Penrose that describes the time evolution of the plasma density. We also compute the resolvent of the Volterra integral operator using the collocation scheme. By integrating a highly oscillatory function that fits in the framework of the first half of the thesis, we are able to reconstruct the velocity distribution of the plasma at any time. The numerical simulations agree with key theoretical aspects of plasma physics such as Landau Damping. We present examples for a Maxwellian distribution and a double-hump unstable distribution.