Skip to main content
SpringerLink
Log in

Parallel Partitioning Strategies for the Adaptive Solution of Conservation Laws

  • Conference paper
Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 75))

Abstract

We describe and examine the performance of adaptive methods for solving hyperbolic systems of conservation laws on massively parallel computers. The differential system is approximated by a discontinuous Galerkin finite element method with a hierarchical Legendre piecewise polynomial basis for the spatial discretization. Fluxes at element boundaries are computed by solving an approximate Riemann problem; a projection limiter is applied to keep the average solution monotone; time discretization is performed by Runge-Kutta integration; and a p-refinement-based error estimate is used as an enrichment indicator. Adaptive order (p-) and mesh (h-) refinement algorithms are presented and demonstrated. Using an element-based dynamic load balancing algorithm called tiling and adaptive p-refinement, parallel efficiencies of over 60% are achieved on a 1024-processor nCUBE/2 hypercube. We also demonstrate a fast, tree-based parallel partitioning strategy for three-dimensional octree-structured meshes. This method produces partition quality comparable to recursive spectral bisection at a greatly reduced cost.

This research was supported by the U.S. Army Research Office Contract Number DAAL03-91-G-0215 and DAALO3-89-C-0038 with the University of Minnesota Army High Performance Computing Research Center (AHPCRC) and the DoD Shared Resource Center at the AHPCRC (Flaherty, Loy); Sandia National Laboratories, operated for the U.S. Department of Energy under Contract Number DE-AC04-76DP00789 (Devine, Wheat), and Research Agreement AD-9585 (Devine); a DARPA Research Assistantship in Parallel Processing administered by the Institute for Advanced Computer Studies, University of Maryland (Loy); and the Grumman Corporate Research Center, Grumman Corporation, Bethpage, NY 11714-3580 (Loy).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 219.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 279.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 279.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Adjerid, M. Aiffa, and J. E. Flaherty, Adaptive Finite Element Methods for Singularly Perturbed Elliptic and Parabolic Systems, submitted for publication, 1993.

    Google Scholar 

  2. S. Adjbrid and J. E. Flaherty, Second–Order Finite Element Approximations and A Posteriori Error Estimation for Two–Dimensional Parabolic Systems, Numer. Math., Vol. 53, 1988, pp. 183–198.

    Article  MathSciNet  Google Scholar 

  3. S. Adjerid. J. Flaherty, P. Moore, and Y. Wang, High–Order Adaptive Methods for Parabolic Systems, Physical–D, Vol. 60, 1992, pp. 94–111.

    MathSciNet  MATH  Google Scholar 

  4. S. Adjerid, J. Flaherty, and Y. Wang, A Posteriori Error Estimation with Finite Element Methods of Lines for One–Dimensional Parabolic Systems, Nu– mer. Math., Vol. 65, 1993, pp. 1–21.

    MathSciNet  MATH  Google Scholar 

  5. D. C. Arney and J. E. Flaherty, An Adaptive Mesh Moving and Local Refinement Method for Time–Dependent Partial Differential Equations, ACM Trans. Math. Software, Vol. 16, 1990, pp. 48–71.

    Google Scholar 

  6. I. Babuska, The p and hp–Versions of the Finite Element Method. The State of the Art, in “Finite Elements: Theory and Appliations,” Springer–Verlag, New York,1988.

    Google Scholar 

  7. I. Babuska, B. A. Szabo, and I. N. Katz, The P– Version Of The Finite Element Method, Si Am J. Numer. Anal., Vol. 18, 1981, pp. 515–545.

    Google Scholar 

  8. M. J. Berger and J. Oliger, Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations, J. Comput. Phys., Vol. 53, 1984. pp. 484–512.

    Google Scholar 

  9. K. S. Bey and J. T. Oden, An A Posteriori Error Estimate for Hyperbolic Condensation Laws, Preprint, 1993.

    Google Scholar 

  10. M. B. Birterman and I. Babuska, The Finite Element Method for Parabolic Equations, II. A Posteriori Error Estimation and Adaptive Approach, Numer. Math., Vol. 40, 1982, pp. 373–406.

    Google Scholar 

  11. R. Biswas, K. D. Devine, and J. E. Flaherty, Parallel, Adaptive Finite Element Methods for Conservation Laws, Appl. Numer. Math.. Vol. 14, 1994, pp. 255– 283.

    Google Scholar 

  12. R. Biswas, J. E. Flaherty, and D. C. Ahnby, An Adaptive Mesh Moving and Refinement Procedure for One–Dimensional Conservation Laws, Appl. Numer. Math., 1993, to appear.

    Google Scholar 

  13. B. Cockburn, S.–Y. Lin, and C.–W. Shu, TVB Rungc–Kutta Local Projection Discontinuous Clalerkin Finite Element Method for Conservation Laws III: One–Dimensional Systems, J. Comput. Phys., Vol. 84, 1989, pp. 90–113.

    Google Scholar 

  14. B. Cockburn and C.W. Shu, TVB Runge–KuttalMcal Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework. Math. Comp., Vol. 52, 1989, pp. 411–435.

    Google Scholar 

  15. B. Cockburn. S.–Y. Lin, and C.–W. Shu, TVB Runge–Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws IV: The Multidimensional Case, Math. Comp.. Vol. 54, 1990, pp. 545–581.

    Google Scholar 

  16. P. Devloo, J. T. Oden, and p. Pattani, An h p Adaptive Finite Element Method for the Numerical Simulation of Compressible Flow, Comput. Method» Appl. Mcch. Engng., Vol. 70, 1988, pp. 203–235.

    Google Scholar 

  17. S. Dey, personal communication, 1993.

    Google Scholar 

  18. M. Dinar, Personal communication, 1993

    Google Scholar 

  19. S. Hammond, Mapping Unstructured Grid Computations to Massively Parallel Computers, Ph.D. Dissertation, Rensselaer Polytechnic Institute, Dept. Comp. Sei., Troy, 1992.

    Google Scholar 

  20. B. Hendrickson and R. Leland, An Improved Spectral Graph Partitioning Algorithm for Mapping Parallel Computations, Sandia National Laboratories Tech. Rep. SAND92–1460, Albuquerque. 1992.

    Google Scholar 

  21. B. Hendrickson and R. Leland, Multidimensional Spectral load Balancing, San–241 dia National Laboratories Tech. Rep. Sand93–0074. Albuquerque, 1993.

    Google Scholar 

  22. Z. Johan, Personal communication, 1993

    Google Scholar 

  23. Z. Johan, K. Mathur, and S. L. Johnsson, An Efficient Communication Strat egy for Finite Element Methods on the Connection Machine CM–5 System, Thinking Machines Tcch. Rep. No. 256, 1993, submitted for publication.

    Google Scholar 

  24. E. Leiss and H. Reddy, Distributed Load Balancing: Design and Performance Analysis, W. M. Keck Research Computation Laboratory, Vol. 5, 1989, pp. 205–270.

    Google Scholar 

  25. R. A. Ludwig, J. E. Flaherty, F. Guerinoni, P. L. Baehmann, and L. S. Shephard, Adaptive Solutions of the Euler Equations Using Finite Quadtree and Octree Grids, Computers and Structures, Vol. 30, 1988, pp. 327–336.

    Google Scholar 

  26. P. K. Moore and J. E. Flaherty, A Local Refinement Finite–Element Method for One–Dimensional Parabolic Systems, Siam J. Numer. Anal., Vol. 27, 1990, pp.1422–1444

    Google Scholar 

  27. A. Pothen, H. Simon, and K.–P. Liou, Partitioning Sparse Matrices with Eigen vectors of Graphs, Siam J. Matrix Analyst* and Applications, Vol. 11, 1990, pp.430–452.

    Google Scholar 

  28. E. Rank and I. Babuska. An Expert System for the Optimal Mesh Design in the hp–Version of the Finite Element Method, Int. J. Numer. Meths. Engng., Vol. 24, 1987, pp.2087–2106.

    Google Scholar 

  29. H. N. Reddy, On Load Balancing, Ph.D. Dissertation, Dept. Comp. Sci., Univ. of Houston. Houston, TX, 1989.

    Google Scholar 

  30. M. S. Shbphard and M. K. Georges, Automatic Three–Dimensional Mesh Generation by the Finite Octree Technique, Int. J. Numer. Meths. Engng., Vol. 32, No. 4, 1991, pp. 709–749.

    Google Scholar 

  31. C.W. Shu and S. Osher. Efficient Implementation of Essentially Non–Oscillatory Shock–Capturing Schemes, II, J. of Comput. Phys., Vol. 27, 1978, pp. 1–31.

    Article  Google Scholar 

  32. H. D. Simon, Partitioning of Unstructured Problems for Parallel Processing, Comput. Systs. Engng., Vol. 2, 1991, pp. 135–148.

    Google Scholar 

  33. P. K. Sweby, High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SIAM J. Numer. Anal., Vol. 21, 1984, pp. 995–1011.

    Article  MathSciNet  MATH  Google Scholar 

  34. B. I. Abo and I. Babuska, Introduction to Finite Element Analysis, j. Wiley and Sons, New York, 1990.

    Google Scholar 

  35. B. Van Leer, Flux Vector Splitting for the Euler Equations, ICASE Report. No. 82–30, Inst. Comp. Applies. Sci. Engng., NASA Langlcy Research Center, Hampton, 1982.

    Google Scholar 

  36. B. Van Leer, Towards the Ultimate Conservative Difference Scheme. IV. A New Approach to Numerical Convection, J. Comput. Phys., Vol. 23, 1977, pp. 276– 299.

    Google Scholar 

  37. S. R. Wheat, A Fine Grained Data Migration Approach to Application Load Balancing on MP MIMD Machines, Ph.D. Dissertation, Dept. Comp. Sci., Univ. of New Mexico, Albuquerque, 1992.

    Google Scholar 

  38. S. R. Wheat, K. D. Devine, and A. B. Maccabe, Experience with Automatic,242 Dynamic Load Balancing and Adaptive Finite Element Computation, Proc. 27th Ann. Hawaii Int. Conf. System Sciences, 1994, pp. 463–472.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Rensselaer Polytechnic Institute, 12180-3590, Troy, NY, USA

    Karen D. Devine

  2. Department of Computer Science and Scientific Computation Research Center, Rensselaer Polytechnic Institute, 12180-3590, Troy, NY, USA

    Joseph E. Flaherty & Raymond M. Loy

  3. Applied Mathematics and Mechanics Section, Benét Laboratories, Watervliet Arsenal, 12189, Watervliet, NY, USA

    Joseph E. Flaherty & Raymond M. Loy

  4. Massively Parallel Computation Research Laboratory, Sandia National Laboratories, 87185-1109, Albuquerque, NM, USA

    Stephen R. Wheat

Authors
  1. Karen D. Devine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph E. Flaherty
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raymond M. Loy
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stephen R. Wheat
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College park, MD, USA

    Ivo Babuska

  2. CIC-3 MS K987 Los Alamos National Laboratory, 87545, Los Alamos, NM, USA

    William D. Henshaw

  3. Research Institute for Advanced Computer Science, NASA Ames Research Center, Mail Stop T20G-5, 94035, Moffet Field, CA, USA

    Joseph E. Oliger

  4. Department of Science and Scientific Computation Research Ctr, Rensselaer Polytechnic Institute, 110-8th St, 12180, Troy, NY, USA

    Joseph E. Flaherty

  5. College of Engineering, Joseph Silbert Dean of Engineering Cornell University Cornell University, 242 Carpenter Hall, 14853, Ithaca, NY, USA

    John E. Hopcroft

  6. Army High Performance Computing Research Center, 1100 South Washington Ave. Suite 101, 55415, Minneapolis, MN, USA

    Tayfun Tezduyar

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag New York

About this paper

Cite this paper

Devine, K.D., Flaherty, J.E., Loy, R.M., Wheat, S.R. (1995). Parallel Partitioning Strategies for the Adaptive Solution of Conservation Laws. In: Babuska, I., Henshaw, W.D., Oliger, J.E., Flaherty, J.E., Hopcroft, J.E., Tezduyar, T. (eds) Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-4248-2_12

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-4248-2_12

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-8707-0

  • Online ISBN: 978-1-4612-4248-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation