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4 Tips for Implementing Multigrid Methods on Domains with Holes

This presentation by Scott Hawley & Richard Matzner at the APS April Meeting in 2003 explores implementing multigrid methods for constrained evolution on domains with holes. The strategies discussed include utilizing 2nd order finite difference methods, incorporating a fast elliptic solver, and overcoming challenges near holes using specialized techniques. Tips such as boundary/restriction schemes, focusing smoothing, utilizing Pre-CGC smoothing, and implementing 'Fake Robin' outer boundary conditions are shared for successful multigrid application. The presentation concludes with future work ideas like black hole solutions, inner BC extrapolation, and parallelization possibilities.

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4 Tips for Implementing Multigrid Methods on Domains with Holes

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  1. 4 Tips for Implementing Multigrid Methods on Domains with Holes Scott Hawley & Richard Matzner Center for Relativity University of Texas at Austin APS April Meeting, 2003

  2. Multigrid for Constrained Ev. • Want to do ‘constrained evolution’ • with excision • Using 2nd order finite difference methods • Need a fast elliptic solver • Try (vertex-centered) multigrid

  3. Sad News: You Won’t Get 2nd O. • We were advised that obtaining 2nd order convergence near holes is impossible or impractical, especially in 3D,… • Unless we… • use ‘fancy’ methods (e.g. deferred correction) • OR use a cubical excision region, having same domain on all multigrid levels

  4. Our Stubbornness • Our application uses Kerr-Schild type data • Analytic, exact inner BC (f=1) • “Should be a simple implementation…” • Didn’t want to be limited to excising only cubes

  5. Our Test Case • Try solving nonlinear Poisson eq: D2 u(x,y,z) +u2(x,y,z) = f(x,y,z) • Inner BC: Dirichlet, using exact value u(x,y,z) • Outer BC: Dirichlet or Robin: [r(u-1)],r=0 • SUCCESS! Get global 2nd O conv.

  6. Tip 1: Boundary/Restriction Scheme • Dirichlet BC on edge of hole (circled X’s) using u(x,y,z) • Use weighted restriction if no neighboring fine grid points are on excision mask, otherwise just “copy” • To anyone: Why does this work? FYI, NOT:

  7. Tip 2: Focus Smoothing Where Needed Error concentrated near holes: Define “Extra Smoothing Region”: Smooth ESR points twice as much as normal points

  8. Results w/ Extra Smoothing Region

  9. Pre- vs. Post-CGC Smoothing Two Holes • Tip 3: Pre-CGC smooths more effective than Post-CGC smooths Note: For multiple V-cycles, you need Post-CGC smooths!

  10. Tip 4: ‘Fake Robin’ Outer BC is OK… • Robin: [r(u-1)],r = 0 • Can just use simple, first order deriv in direction normal to faces of cubical domain [Alcubierre]

  11. Wrapping Up • 4 Tips: • Put boundary points at edge of hole, but don’t use them in weighted restriction • Use Extra Smoothing Region • Try more Pre- than Post-CGC smooths • For Robin BC: Use first-order perp. deriv • Future Work: • Black hole solutions! • Need to extrapolate for inner BC? • Parallelization

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