Replica time integrators

Abstract

SUMMARYThis paper is concerned with the classical problem of wave propagation in discrete models of nonuniform spatial resolution. We develop a new class of Replica Time Integrators (RTIs) that permit the two-way transmission of thermal phonons across mesh interfaces. This two-way transmissibility is accomplished by representing the state of the coarse regions by means of replica ensembles, consisting of collections of identical copies of the coarse regions. In dimension d, RTIs afford an O(n d ) speed-up factor in sequential mode, and O(n d+1 ) in parallel, over regions that are coarsened n-fold. In this work, we restrict ourselves to the solution of the 3d continuous wave equation, for both linear and non-linear materials. By a combination of phase-error analysis and numerical testing, we show that RTIs are convergent and result in exact two-way transmissibility at the Courant-Friedrichs-Lewy limit for any angle of incidence. In this limit, RTIs allow step waves and high-frequency harmonics to cross mesh interfaces in both directions without internal reflections or appreciable loss or addition of energy. The possible connections of RTIs with discrete-to-continuum approaches and, in particular, with the transition between molecular dynamics and continuum thermodynamics are also pointed to by way of future outlook.

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

“…Improvements of the method in the future work to further reduce the computational cost may include applying the fast multipole method for solving the elastic field [43,44] or using locally varying time stepping [36][37][38].…”

“…The values of the level set functions on the coarse grid provide rough estimates of them, which are essential for the reconstruction of the fine mesh during the evolution of the high codimensional objects as discussed earlier. Moreover, this two‐level adaptive method provides a basis for using locally varying time stepping to further reduce the computational cost. In this paper, we present this method and the simulation results on a two‐level adaptive mesh with uniform grids for moving curves in two and three dimensions and for dislocation dynamics in three dimensions.…”

Absorbing Interface Conditions for the Simulation of Wave Propagation on NonUniform Meshes

“…where wave speed c = 372.82 Å/ps, a 1 = 6.17 × 10 −3 , a 2 = 1 × 10 −2 , a 3 = 1 × 10 −1 . The initial value is taken to zero with a displacement boundary condition in [29]. There are four different frequency wave in the waves, of which the solution on a uniform fine mesh has been plotted in Fig.…”

“…Replica time integration (RTI) [29,33], on the other hand, also plays a significant role recently in this field. It builds up an interface condition that allows two-way transmission for high-frequency waves.…”