Replica time integrators

Abstract

This 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.