Efficient Parallel Computing for Laser-Gas Quantum Interaction and Propagation E. Lorin University of Ontario Institute of Technology Faculty of Science 2000, Simcoe North Street, L1H 7K4, Oshawa, Canada Emmanuel.Lorin@uoit.ca A. D. Bandrauk Laboratoire de chimie th´ eorique Facult´ e des Sciences Universit´ e de Sherbrooke, Qu´ e, J1K 2R1, Canada Andre.Bandrauk@USherbrooke.ca Abstract We present in this short paper some domain decomposi- tion and high performance numerical techniques for simu- lating laser-gas interaction and propagation using a multi- scale Maxwell-Schr¨ odinger model. Ionization and high or- der harmonic generation are in particular taken into ac- count in this micro-macro model, leading to additional sim- ulation difficulties. We propose some benchmarks to vali- date the presented techniques. 1. Introduction Simulation of intense and ultrashort laser-matter interac- tion necessitates a very precise micro-macro modeling. In [4], [5] we have introduced a multi-scale model coupling the macroscopic Maxwell equations modeling the propagation of an electro-magnetic field in a gas modeled by many Time Dependent Schr ¨ odinger Equations (TDSE’s). The complex- ity of this model especially in 3-D, requires the use of effi- cient techniques. In this paper, we present some numer- ical and parallel methods for simulating efficiently ultra- short (less than 10 -14 second), intense (more than 10 13 W · cm -2 ) and high frequency (less than 800 nm) laser pulses propagating in dense (more than 10 17 mol · cm -3 ) gaseous media. At this point, numerical schemes used are still rela- tively standard and consists of the use of a Yee-like scheme for the Maxwell equations and Crank-Nicolson scheme for the TDSE’s and multi-grid techniques. As discussed later in the paper the TDSE approximation is the most costly part of the simulation due the fact that an accurate description of the gas requires the computation of hundreds or thousands TDSE’s. In Sections 2, 3 we present shortly the Maxwell- Schr¨ odinger model and its numerical approximation. We will consider the case of non Born-Oppenheimer approxi- mations (moving nuclei) even if most of the computations are done under the Born-Oppenheimer approximation al- lowing to reduce the complexity of the problem. We focus in particular on the boundary conditions for laser-molecule TDSE’s (still in Section 3) allowing a crucial reduction of the algorithmic complexity of the numerical scheme ap- proximating these equations. Then a domain decomposition approach is presented and its natural parallelization in Sec- tion 4. A numerical experiment is then proposed to validate the chosen method. 2 Physical problem and modeling The model we consider is a coupling between the mi- croscopic laser-molecule Schr¨ odinger equations and macro- scopic Maxwell’s equations. It allows to take into account high order harmonic generation, ionization and is then more precise than classical nonlinear Schr¨ odinger’s equations. Some informations related to the model may be found in [5] and some applications in [6] or [7]. As it is not the purpose of this paper we simply present the system of equations (in atomic unit). First, we introduce some important notations. For the Maxwell equations, we will denote by Ω ⊂ R 3 the spatial domain with a regular boundary denoted by Γ and by r =(x, y, z) the space variable in Ω. At the molecule scale we will denote by ( r ′ =(x ′ ,y ′ ,z ′ ),R ′ ) ∈ R 3 × R ∗ + the space variable (for electrons and ions). The molecular 22nd International Symposium on High Performance Computing Systems and Applications (HPCS 2008) 1550-5243/08 $25.00 © 2008 IEEE DOI 10.1109/HPCS.2008.14 1 22nd International Symposium on High Performance Computing Systems and Applications (HPCS 2008) 1550-5243/08 $25.00 © 2008 IEEE DOI 10.1109/HPCS.2008.14 4