Time exponential splitting integrator for the Klein–Gordon equation with free parameters in the Hagstrom–Warburton absorbing boundary conditions

The Klein–Gordon equation on an infinite two dimensional strip is considered. Numerical computation is reduced to a finite domain by using the Hagstrom–Warburton (H–W) absorbing boundary conditions (ABCs) with free parameters in the formulation of the auxiliary variables. The spatial discretization is achieved by using fourth order finite differences and the time integration is made by means of an efficient and easy to implement fourth order exponential splitting scheme which was used in Alonso-Mallo and Portillo (2016) considering the fixed Padé parameters in the formulation of the ABCs. Here, we generalize the splitting time technique to other choices of the parameters. To check the timeintegrator we consider, on one hand, fourty peso ffixed parameters, the Newmann’s parameters, the Chebyshev’s parameters, the Padé’s parameters and optimal parameters proposed in Hagstrom et al. (2007) and, on the other hand, an adaptive scheme for the dynamic control of the order of absorption and the parameters. We study the efficiency of the splitting scheme by comparing with thefourth-order four-stage Runge–Kutta method.MTM2015-66837-P del Ministerio de Economía y Competitivida

Three-dimensional Quantum Slit Diffraction and Diffraction in Time

We study the quantum slit diffraction problem in three dimensions. In the treatment of diffraction of particles by a slit, it is usually assumed that the motion perpendicular to the slit is classical. Here we take into account the effect of the quantum nature of the motion perpendicular to the slit using the Green function approach [18]. We treat the diffraction of a Gaussian wave packet for general boundary conditions on the shutter. The difference between the standard and our three-dimensional slit diffraction models is analogous to the diffraction in time phenomenon introduced in [16]. We derive corrections to the standard formula for the diffraction pattern, and we point out situations in which this might be observable. In particular, we discuss the diffraction in space and time in the presence of gravity

Near conserving energy numerical schemes for two-dimensional coupled seismic wave equations

Two-dimensional coupled seismic waves, satisfying the equations of linear isotropic elasticity, on a rectangular domain with initial conditions and periodic boundary conditions, are considered. A quantity conserved by the solution of the continuous problem is used to check the numerical solution of the problem. Second order spatial derivatives, in the x direction, in the y direction and mixed derivative, are approximated by finite differences on a uniform grid. The ordinary second order in time system obtained is transformed into a first order in time system in the displacement and velocity vectors. For the time integration of this system, second order and fourth order exponential splitting methods, which are geometric integrators, are proposed. These explicit splitting methods are not unconditionally stable and the stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.MTM2015-66837-P del Ministerio de Economía y Competitivida

Photoabsorption spectra in the continuum of molecules and atomic clusters

We present linear response theories in the continuum capable of describing photoionization spectra and dynamic polarizabilities of finite systems with no spatial symmetry. Our formulations are based on the time-dependent local density approximation with uniform grid representation in the three-dimensional Cartesian coordinate. Effects of the continuum are taken into account either with a Green's function method or with a complex absorbing potential in a real-time method. The two methods are applied to a negatively charged cluster in the spherical jellium model and to some small molecules (silane, acetylene and ethylene).Comment: 13 pages, 9 figure

A k-space method for nonlinear wave propagation

A k-space method for nonlinear wave propagation in absorptive media is presented. The Westervelt equation is first transferred into k-space via Fourier transformation, and is solved by a modified wave-vector time-domain scheme [Mast et al., IEEE Tran. Ultrason. Ferroelectr. Freq. Control 48, 341-354 (2001)]. The present approach is not limited to forward propagation or parabolic approximation. One- and two-dimensional problems are investigated to verify the method by comparing results to the finite element method. It is found that, in order to obtain accurate results in homogeneous media, the grid size can be as little as two points per wavelength, and for a moderately nonlinear problem, the Courant-Friedrichs-Lewy number can be as small as 0.4. As a result, the k-space method for nonlinear wave propagation is shown here to be computationally more efficient than the conventional finite element method or finite-difference time-domain method for the conditions studied here. However, although the present method is highly accurate for weakly inhomogeneous media, it is found to be less accurate for strongly inhomogeneous media. A possible remedy to this limitation is discussed

Elliptic harbor wave model with perfectly matched layer and exterior bathymetry effects

Standard strategies for dealing with the Sommerfeld condition in elliptic mild-slope models require strong assumptions on the wave field in the region exterior to the computational domain. More precisely, constant bathymetry along (and beyond) the open boundary, and parabolic approximations–based boundary conditions are usually imposed. Generally, these restrictions require large computational domains, implying higher costs for the numerical solver. An alternative method for coastal/harbor applications is proposed here. This approach is based on a perfectly matched layer (PML) that incorporates the effects of the exterior bathymetry. The model only requires constant exterior depth in the alongshore direction, a common approach used for idealizing the exterior bathymetry in elliptic models. In opposition to standard open boundary conditions for mild-slope models, the features of the proposed PML approach include (1) completely noncollinear coastlines, (2) better representation of the real unbounded domain using two different lateral sections to define the exterior bathymetry, and (3) the generation of reliable solutions for any incoming wave direction in a small computational domain. Numerical results of synthetic tests demonstrate that solutions are not significantly perturbed when open boundaries are placed close to the area of interest. In more complex problems, this provides important performance improvements in computational time, as shown for a real application of harbor agitation.Peer ReviewedPostprint (author's final draft