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PAMR/AMRD Overview

PAMR/AMRD Overview. FLAMR II Workshop, CITA May 24 th , 2006. Outline. properties of the astrophysical scenarios we are interested in exploring mathematical structure physical properties computational techniques model problem

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PAMR/AMRD Overview

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  1. PAMR/AMRD Overview FLAMR II Workshop, CITA May 24th, 2006

  2. Outline • properties of the astrophysical scenarios we are interested in exploring • mathematical structure • physical properties • computational techniques • model problem • Berger and Oliger style adaptive mesh refinement for coupled elliptic/hyperbolic systems • PAMR/AMRD — a computational hierarchy for implementing B&O AMR using finite difference techniques • extending PAMR to support cell-centered grid structures and solution methods • overview of design • what still needs to be done before implementation of model problem can begin

  3. Astrophysical Scenarios • of immediate interest are dynamical astrophysical events where strong-field gravity is expected to play a significant role, in particular compact object coalescence: • black hole/black hole, neutron star/neutron star, neutron star/black hole • interested in the gravitational waves that will be emitted by such events • for black holes, general relativity (GR) provides all the physics • for neutron stars, will need to build towards a realistic description of its material structure • first cut for matter : single component fluid with approximate nuclear EOS • most important aspect for studying gravitational waves will be to resolve the bulk motion of matter

  4. Mathematical Structure of Equations • GR • much freedom in choosing the coordinates/system of variables/formalisms/etc. to achieve a particular mathematical structure • free evolution : hyperbolic PDEs • constrained evolution : hyperbolic and elliptic PDEs • characteristic (null) evolution: hyperbolic PDEs and ODEs • in all cases equations are coupled and non-linear • hydrodynamics and E&M • in most cases hyperbolic systems, on occasion elliptics (eq. “divergence-cleaning” in E&M)

  5. Computational Techniques • GR – we use vertex-centered finite difference discretization • Newton-Gauss-Seidel relaxation for hyperbolic PDEs • FAS (full approximation storage) multigrid for elliptics • Hydrodynamics plus E&M • that’s the goal of this workshop! • most common methods based on flux conservative and “high resolution” shock capturing schemes, which for finite difference discretization typically requires cell-centered methods • In situations of interest the matter & geometry will be highly coupled and span several orders of magnitude of relevant spatial and temporal length scales • to resolve the length scales we need parallel adaptive mesh refinement (AMR) • strong coupling implies that all fields should be adequately resolved on identical grid hierarchies, at least to study gravitational wave emission

  6. Simplified Model Problem • Description

  7. Berger and Oliger AMR • (simplified and extended) Berger and Oliger AMR • computational domain covered by a hierarchy of rectangular meshes, where higher resolutionchildmeshes are aligned with and entirely contained within coarser resolution parent meshes • adequate for the kinds of problems we’re interested in, and convenient for a straightforward implementation of multigrid • key features of the algorithm • hierarchy construction driven by truncation error estimates • refinements occur in space and time (i.e. time-subcycling) • gives optimal O(N) solution of the class of PDEs we solve • algorithm extended to incorporate elliptic PDEs • method is expected to work for coupled systems of equations where the dynamics of the solution are driven by the subset of hyperbolic fields • in theory is transparent to the user, i.e. evolution proceeds via a series of unigrid time steps

  8. B&O AMR time stepping • Recursive time stepping • a single time step on a parent level is taken before rt (temporal refinement ratio) steps are taken on the child level • Again, reason for time sub-cycling is to get solution in O(N); however if we are going to do time sub-cycling the above recursion is crucial to set boundary conditions for interior equations, in particular the elliptics • alternative strategies possible for purely hyperbolic systems with explicit time integration, or certain classes of linear elliptic PDEs driven by conserved sources

  9. B&O AMR Example time 2 Levels = 2:1 = 2:1 Next frame: Level 1evolution time step

  10. B&O AMR Example time 2 Levels = 2:1 = 2:1 Next frame: Level 2evolution time step

  11. B&O AMR Example time 2 Levels = 2:1 = 2:1 Next frame: Level 2evolution time step

  12. B&O AMR Example time 2 Levels = 2:1 = 2:1 Next frame: Injectionfrom level 2 to 1

  13. B&O AMR Example time 2 Levels = 2:1 = 2:1 Next frame: Regrid

  14. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 1evolution time step

  15. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 2evolution time step

  16. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 3evolution time step

  17. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 3evolution time step

  18. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Injectionfrom level 3 to 2

  19. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 2evolution time step

  20. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Injectionfrom level 2 to 1

  21. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 3evolution time step

  22. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Level 3evolution time step

  23. B&O AMR Example time 3 Levels = 2:1 = 2:1 Next frame: Injectionfrom level 3 to 2

  24. B&O AMR Example time 3 Levels = 2:1 = 2:1

  25. PAMR user code 1 AMRD user code 2 . . . user code N PAMR/AMRD • PAMR (parallel adaptive mesh refinement) manages distributed B&O style grid hierarchies • AMRD (adaptive mesh refinement driver) implements a standard version of B&O AMR, utilizing PAMR for hierarchy management • User codes designed as (in-principle) standalone unigrid/serial numerical solvers, and supply AMRD with a series of “hook functions” to incorporate them into the B&O algorithm • Reasons for this separation of functionality • from the point of view of a user writing a code to numerically solve a particular system of PDEs, AMR and parallel distribution are largely extraneous details • all the user should be aware of is the possibility that the code could be run in a parallel/adaptive environment, meaning grid boundaries could either be at the physical boundaries of the problem, or interior to the domain • in the latter case the user leaves the boundaries alone • The AMR driver does not need to know the details of how grids will be distributed in parallel, nor what equations the user will be solving on those grids • PAMR handles the non-local aspects of parallel grid distribution, and does not care what the underlying programs will do with the grids

  26. Basic Functionality • PAMR • AMRD • Newtonian boson star sample code • parameter file • AMRD hook functions • unigrid update routines

  27. Planned support for coupled vertex/cell centered solution methods • Grid hierarchy will support both cell centered (CC) and vertex centered (VC) variables • for each region of the computational domain require both CV and CC grids to occupy the same physical volume • should allow for efficient representation of solutions to systems of PDEs in the tightly coupled regime, i.e. where the locations of relevant small length scale features of the solution are similar for all the fields • a given variable can have both representations; the primary representation will be used to evolve the variable, and the “conjugate” representation will be for use as a source function in equations solved for in this opposite representation • Use a naming convention to identify the type of a variable and its conjugate pair

  28. PAMR to-do list • Internal “machinery” to handle CC representation • Add CC versions of required grid-to-grid communication operations • coarse to fine level interpolation • fine to coarse level injection (averaging) • ghost zone synchronization • question : what are the common interpolation/injection operators used in hydro schemes? • Add operations to refresh conjugate representations of variable: • CC -> VC • VC -> CC • All the above with 1,2 and 3 spatial dimensions

  29. AMRD to-do list • Call PAMR VC->CC and CC->VC at appropriate times • Special ways of handling CC internal boundaries?? • Support for explicit time steppers (in particular Runge-Kutta)

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