Generalized wave propagation problems and discrete exterior calculus

Abstract

We introduce a general class of second-order boundary value problems unifying application areas such as acoustics, electromagnetism, elastodynamics, quantum mechanics, and so on, into a single framework. This also enables us to solve wave propagation problems very efficiently with a single software system. The solution method precisely follows the conservation laws in finite-dimensional systems, whereas the constitutive relations are imposed approximately. We employ discrete exterior calculus for the spatial discretization, use natural crystal structures for three-dimensional meshing, and derive a “discrete Hodge” adapted to harmonic wave. The numerical experiments indicate that the cumulative pollution error can be practically eliminated in the case of harmonic wave problems. The restrictions following from the CFL condition can be bypassed with a local time-stepping scheme. The computational savings are at least one order of magnitude.

Three-dimensional splitting dynamics of giant vortices in Bose-Einstein condensates

“…The third option is the C15 structure, which is one of the tetrahedrally closepacked tilings [57][58][59][60]. The C15 structure has been found to be a high-quality grid for the solution of the Maxwell equations [47,53]. For each of these three grid types, we employ three discretization levels, where tasks are scaled to involve 10 9 , 10 10 , or 10 11 floating point multiplications for integration over a unit time interval.…”

“…The time span of stability appears to be very sensitive to the grid type used. The BCC grid offers the longest time spans, since it is numerically the most isotropic of the three grids [53]. With the finest discretization level, BCC leads to threefold splitting, which is the most likely physical solution for the used parameter values (see Sec.…”

General conservation law for a class of physics field theories

“…The term "discrete Hodge" is just a compound word.) Examples of applying Yee-like schemes to the general conservation law can be found in [24].…”

Discrete exterior calculus for photonic crystal waveguides

“…The DEC enables such grid structures that can handle complex domains. Since only inversion of a diagonal matrix is required at each timestep, it also provides efficient time‐stepping 10 . The method is pioneered in electromagnetic simulations by Bossavit and Kettunen, 11 while the groundwork presented by Marsden's group was applied in computer vision and image processing 12 and delivered the PyDEC software library 13 .…”

“…Despite the wide number of considered application areas and numerical experiments there exist only a few examples of wave propagation in heterogeneous domain in which the time‐dependent problem is discretized with the DEC. These include electromagnetic scattering by objects such as hexagonal ice crystals or spherical particles randomly distributed into the surrounding media 10,22 . In the ice crystal simulations, the object has been considered to be large compared to the wavelength, while spherical particles were small compared to the wavelength.…”