ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation
Abstract
Resolving numerically Vlasov–Poisson equations for initially cold systems can be reduced to following the evolution of a threedimensional sheet evolving in sixdimensional phasespace. We describe a public parallel numerical algorithm consisting in representing the phasespace sheet with a conforming, selfadaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six and fourdimensional phasespace. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phasespace sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65–67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phasespace the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuationmore »
 Authors:
 Institut d'Astrophysique de Paris, CNRS UMR 7095 and UPMC, 98bis, bd Arago, F75014 Paris (France)
 (Japan)
 Publication Date:
 OSTI Identifier:
 22572354
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 321; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CHAOS THEORY; COSMOLOGY; EQUATIONS OF MOTION; HAMILTONIANS; LAGRANGIAN FUNCTION; NONLUMINOUS MATTER; PHASE SPACE; POISSON EQUATION; POTENTIALS; SHEETS; SIMULATION
Citation Formats
Sousbie, Thierry, Email: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 1130033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 1130033, Colombi, Stéphane, Email: colombi@iap.fr, and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 6068502. ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.05.048.
Sousbie, Thierry, Email: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 1130033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 1130033, Colombi, Stéphane, Email: colombi@iap.fr, & Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 6068502. ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation. United States. doi:10.1016/J.JCP.2016.05.048.
Sousbie, Thierry, Email: tsousbie@gmail.com, Department of Physics, The University of Tokyo, Tokyo 1130033, Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 1130033, Colombi, Stéphane, Email: colombi@iap.fr, and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 6068502. 2016.
"ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation". United States.
doi:10.1016/J.JCP.2016.05.048.
@article{osti_22572354,
title = {ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation},
author = {Sousbie, Thierry, Email: tsousbie@gmail.com and Department of Physics, The University of Tokyo, Tokyo 1130033 and Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 1130033 and Colombi, Stéphane, Email: colombi@iap.fr and Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 6068502},
abstractNote = {Resolving numerically Vlasov–Poisson equations for initially cold systems can be reduced to following the evolution of a threedimensional sheet evolving in sixdimensional phasespace. We describe a public parallel numerical algorithm consisting in representing the phasespace sheet with a conforming, selfadaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six and fourdimensional phasespace. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phasespace sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65–67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phasespace the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a “warm” dark matter simulation in sixdimensional phasespace that we use to check the parallel scaling of the code.},
doi = {10.1016/J.JCP.2016.05.048},
journal = {Journal of Computational Physics},
number = ,
volume = 321,
place = {United States},
year = 2016,
month = 9
}

We present a direct, adaptive solver for the Poisson equation which can achieve any prescribed order of accuracy. It is based on a domain decomposition approach using local spectral approximation, as well as potential theory and the fast multipole method. In two space dimensions, the algorithm requires O(NK) work, where N is the number of discretization points and K is the desired order of accuracy. 32 refs., 6 figs., 4 tabs.

AFMPB: An adaptive fast multipole Poisson Boltzmann solver for calculating electrostatics in biomolecular systems
A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized PoissonBoltzmann equation. The numerical solver utilizes a wellconditioned boundary integral equation (BIE) formulation, a nodepatch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipoletolocal translations. The program and its full description, as well as several closely related libraries and utility tools are available at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: themore » 
iAPBS: a programming interface to Adaptive PoissonBoltzmann Solver
The Adaptive PoissonBoltzmann Solver (APBS) is a stateoftheart suite for performing PoissonBoltzmann electrostatic calculations on biomolecules. The iAPBS package provides a modular programmatic interface to the APBS library of electrostatic calculation routines. The iAPBS interface library can be linked with a Fortran or C/C++ program thus making all of the APBS functionality available from within the application. Several application modules for popular molecular dynamics simulation packages  Amber, NAMD and CHARMM are distributed with iAPBS allowing users of these packages to perform implicit solvent electrostatic calculations with APBS. 
Mathematical and Numerical Aspects of the Adaptive Fast Multipole PoissonBoltzmann Solver
This paper summarizes the mathematical and numerical theories and computational elements of the adaptive fast multipole PoissonBoltzmann (AFMPB) solver. We introduce and discuss the following components in order: the PoissonBoltzmann model, boundary integral equation reformulation, surface mesh generation, the nodepatch discretization approach, Krylov iterative methods, the new version of fast multipole methods (FMMs), and a dynamic prioritization technique for scheduling parallel operations. For each component, we also remark on feasible approaches for further improvements in efficiency, accuracy and applicability of the AFMPB solver to largescale longtime molecular dynamics simulations. Lastly, the potential of the solver is demonstrated with preliminary numericalmore »