Replica time integrators

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(nd ) speed-up factor in sequential mode, and O(nd+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. Copyright 2011 John Wiley & Sons, Ltd.