Progress in My Adaptive Mesh Refinement Technology

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Progress in (My) Adaptive Mesh Refinement
Technology

• Scott H. Hawley
• Albert Einstein Institute, Golm, Germany

• Collaborators
• Manish Parashar, TASSL, Rutgers
• Matthew W. Choptuik, UBC
• Ralf Kaehler, AEI ZIB
• KDI Project WashU (Suen, Evans, Iyer), NCSA
  (Pak-Shing, Shalf), Rutgers, Chicago

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This Project

• Create a general driver whereby users supply
  sequential unigrid code and get parallel AMR code
  ...via Berger Oliger AMR (see next slide)
• Test problem 2D Wave Equation (flat space)
• No point solving harder systems yet
• Use GrACE, successor of DAGH
• Standalone application, to be used as template
  for incorporation into RNPL and/or Cactus
• When will this be in Cactus?
• Answer A When AMR is working reliably.
• Answer B Ask WashU group.
• Answer C Soon.

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Berger Oliger AMR
Berger Oliger, J. Comp. Phys. 53, 1984

• AMR via multiple, uniform rectangular grids w/
  different resolutions
• Grids are distinct domains, made to communicate
  along boundaries, via prolongation
  (interpolation) restriction
• Recursive algorithm, evolve level L0 (coarsest)
  first, call again for level L1...

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Minimal Berger Oliger

• In original BO AMR, fine grids can have
  arbitrary orientation rel. to coarse grids
• But we force fine grid boundaries to lie along
  coarse grid cell boundaries
• Furthermore, we force finer grids to be
  contained inside coarser grids
• ...This introduces a notion of parent (coarser)
  and child (finer) grids

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Truncation Error Estimation

• Bad Points Where TEgtThreshold
• Measure TE by evolving same data on a coarser
  grid (e.g. factor of 2 coarser)

• To avoid overhead, we use a shadow hierarchy
  (Choptuik 1995) exact copy of main hierarchy,
  but coarser, evolved independently
• Main hierarchy grids reqd to lie on boundaries
  of coarse grid cells of the shadow hierarchy

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Berger Rigoutsos Clustering
BR, IEEE Trans. Systems, Man Cybernetics 21,
1991

• How to cover bad points...
• Splits a given grid into pieces
• Decides where to split using edge detection
  algorithm
• Parameters
• (Minimum) Block Size
• Buffer Width
• (Minimum) Efficiency
• Only a unique implementation for AMR with 2
  levels of refinement...

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What I Can Deliver Right Now

• Fixed Mesh Refinement (FMR)
• 2-Level AMR (w/o restriction)
• 2nd-order convergence except at adaptive
  boundaries
• Note that convergence preserved when waves emerge
  on other side of adaptive boundaries
• For better convergence
• higher-order interpolation (e.g., Hearns work,
  Schnetters Carpet)
• Use same physical boundarywidth on Main Shadow?

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BR Clustering, contd

• Options for 3 or more levels
• Feed in global structure (coords of all flagged
  points for all grids at level L)
• can yield efficient clustering
• - can result in children w/ multiple parents
• - may not guarantee containment
• Call clusterer once per grid ()

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Nesting

• Boundaries of level Lgt0 grids are obtained via
  interpolation from coarser grids
• Want new fine grids to lie entirely interior to
  and share no adaptive boundaries with
  underlying coarse grids
• Trick dont flag points in boundary region
• But clusterer can still get confused...

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The Corner Problem
Level 1
X Flagged pts
Proposed Level 2

• Happens in sims where Efficiency isnt very high
• Good news For physics sims where for which AMR
  is valid (TE obeys Richardson...), C.P. shouldnt
  arise if BufferWidth is large enough
• Since user may not know BufferWidth a priori
• User must re-run w/ larger BufferWidth
  (Screwed)
• Have algorithm automatically guarantee nesting...

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Guaranteed Nesting

• 5 Options
• Regrid Everything If nesting problem arises,
  grow underlying coarse grids
• Could propagate info faster than c, even into
  past
• Genius Clusterer Always does what you want
• Cluster-No Accepts forbidden points in addition
  to flagged and allowed points
• TakenBreak Break grids at boundary-type
  transitions
• By Hand / Tweaky Allow multiple TE thresholds
  for different levels user re-runs w/ different
  thresholds

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6th Option Cut Boundaries

• For each grid given to clusterer, split into 5
  sub-grids, by cutting at boundaries
• BR clusterer will naturally cut out non-Xs
  along boundaries
• Can generate extra tiny grids
• Some grids can be joined afterwards

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Further Work

• Regrid Everything perhaps not ideal
• Implement new Cut-Boundaries scheme
• Fix TE management / Restriction operations
• May want to write in-house, simple AMR
  libraries (e.g. Schnetters Carpet My
  clusterer)
• Adaptive load balancing?
• Elliptic solver?