An efficient linearly-implicit energy-preserving scheme with fast solver for the fractional nonlinear wave equation

The paper considers the Hamiltonian structure and develops efficient energy-preserving schemes for the nonlinear wave equation with a fractional Laplacian operator. To this end, we first derive the Hamiltonian form of the equation by using the fractional variational derivative and then applying the finite difference method to the original equation to obtain a semi-discrete Hamiltonian system. Furthermore, the scalar auxiliary variable method and extrapolation technique is used to approximate a semi-discrete system to construct an efficient linearly-implicit energy-preserving scheme. A fast solver for the proposed scheme is presented to reduce CPU consumption. Ample numerical results are given to finally confirm the efficiency and conservation of the developed scheme

Numerical solution of the Klein-Gordon equation in an unbounded domain.

Master of Science in Applied Mathematics. University of KwaZulu-Natal, Westville, 2018.Abstract available in PDF file

A symmetric low-regularity integrator for the nonlinear Schr\"odinger equation

We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schr\"odinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb{T}^d$ and on a smooth bounded domain $\Omega \subset \mathbb{R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e. under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes

Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations

This is the first paper in a series where we study collisions of nucleated bubbles taking into account the effects of small initial (quantum) fluctuations in a fully 3+1-dimensional setting. In this paper, we consider the evolution of linear fluctuations around highly symmetric though inhomogeneous backgrounds. We demonstrate that a large degree of asymmetry develops over time from tiny fluctuations superposed upon planar and SO(2,1) symmetric backgrounds. These fluctuations arise from zero-point vacuum oscillations, so excluding them by enforcing a spatial symmetry is inconsistent in a quantum treatment. We consider the limit of two colliding planar walls, with fluctuation mode functions characterized by the wavenumber transverse to the collision direction and a longitudinal shape along the collision direction $x$, which we solve for. Initially, the fluctuations obey a linear wave equation with a time- and space-dependent mass $m_{eff}(x,t)$. When the walls collide multiple times, $m_{eff}$ oscillates in time. We use Floquet theory to study the fluctuations and generalize techniques familiar from preheating to the case with many coupled degrees of freedom. This inhomogeneous case has bands of unstable transverse wavenumbers $k_\perp$ with exponentially growing mode functions. From the detailed spatial structure of the mode functions in $x$, we identify both broad and narrow parametric resonance generalizations of the homogeneous $m_{eff}(t)$ case of preheating. The unstable $k_\perp$ modes are longitudinally localized, yet can be described as quasiparticles in the Bogoliubov sense. We define an effective occupation number to show they are created in bursts for the case of well-defined collisions in the background. The transverse-longitudinal coupling accompanying nonlinearity radically breaks this localized particle description, with nonseparable 3D modes arising.Comment: 37 pages + references, 20 figures, submitted to JCA