Optimized high-order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions

Journal of Scientific Computing

Volumn 32, Issue 1, 2007, Pages 109-145

  Diener, Peter ^a   Dorband, Ernst Nils ^a   Schnetter, Erik ^a,b   Tiglio, Manuel ^a  

^a LOUISIANA STATE UNIVERSITY   (United States)

^b ALBERT EINSTEIN INSTITUT   (Germany)

Author keywords

Accuracy; Artificial dissipation; High order finite differencing; Multi block evolutions; Numerical stability

Indexed keywords

BOUNDARY POINTS; DISSIPATION OPERATORS; TRUNCATION ERROR;

APPROXIMATION THEORY; BANDWIDTH; BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; OPTIMIZATION;

MATHEMATICAL OPERATORS;

EID: 34250840018     PISSN: 08857474     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10915-006-9123-7     Document Type: Article

References (40)

1. Kreiss, H. O., and Scherer, G., (1974). Finite element and finite difference methods for hyperbolic partial differential equations. In Boor C. D. (ed.), Mathematical Aspects of Finite Elements in Partial Differential Equations, Academica Press, New York.
2. Kreiss, H. O., and Scherer, G. (1977). Tech. Rep., Dept. of Scientific Computing, Uppsala University Sweden.
3. Olsson, P. (1995). Summation by Parts, Projections, and Stability I. Math. Camp. 64, 1035.
4. Olsson, P. (1995). Supplement to "Summation by parts, projections, and stability, I" Math. Comp. 64, S23.
5. Olsson, P. (1995). Summation by Parts, Projections, and Stability II. Math. Comp. 64, 1473.
6. Carpenter, M., Gottlieb, D., and Abarbanel, S. (1994). Time-Stable boundary conditions for finite-difference schemes solving hyperbolic systems. Methodology and application to high-order compact schemes. J. Comput. Phys. 111, 220.
7. Gustafsson, B. (1998). On the implementation of boundary conditions for the method of lines. BIT 38, 293.
8. Mattsson, K. (2003). Boundary procedures for summation by parts operators. J. Sci. Comput. 18, 133.
9. Strand, B. (1996). Ph.D. thesis, Uppsala University, Department of Scientific Computing, Uppsala University. Uppsala, Sweden.
10. Carpenter, M., Nordström, J., and Gottlieb, D. (1999). A Stable and Conservative Interface Treatment of Arbitrary Spatial Accuracy. J. Comput. Phys. 148, 341.
11. Nordström, J., and Carpenter, M. (2001). High-order finite difference methods, multidimensional linear problems and curvilinear coordinates. J. Comput. Phys. 173, 149.
12. Reula, O. (1998). Living Rev. Rel. 1, 3. URL http://relativity.livingreviews.org/Articles/lrr-1998-3/index.html.
13. Rauch, J. (1985). Symmetric positive systems with boundary characteristics of constant multiplicity. Trans. Am. Math. Soc. 291, 167
14. Secchi, P. (1996). The initial boundary value problem for linear symmetric hyperbolic systems with characteristic boundary of constant multiplicity. Differential Integral Equations 9, 671.
15. Secchi, P. (1996). Well-posedness of characteristic symmetric hyperbolic systems. Arch. Rat. Mach. Anal. 134, 155.
16. Lehner, L., Reula, O., and Tiglio, M. (2005). Multi-block simulations in general relativity: high order discretizations, numerical stability, and applications. Class. Quantum Grav. 22 gr-qc/0507004.
17. Strand, B. (1994). Summation by parts for finite differencing approximations for d/dx. J. Comput. Phys. 110, 47.
18. Svärd, M., Mattsson, K., and Nordström, J. (2005). Steady State Computations Using Summation by Parts Operators. J. Sci. Comput. 24, 79.
19. Mattsson, K., Svärd, M., and Nordström, J. (2004). Stable and accurate artificial dissipation. J. Sci. Comput. 21, 57.
20. Friedrich, H. (2002). Conformal Einstein evolution. Lect. Notes Phys. 604, 1, gr-qc/0209018.
21. Tadmor, E. (1994). Spectral methods for hyperbolic problems. In Lecture notes delivered at Ecole des Ondes, "Méthodes numériques d'ordre élevé pour les ondes en régime transitoire", INRIA-Rocquencourt January 24-28. URL http://www.cscamm.umd.edu/people/faculty/tadmor/pub/spectral-approximations/ Tadmor.INRIA-94.pdf.
22. Svärd, M. (2004). On coordinate transformations for summation-by-parts operators. J. Sci. Comput. 20, 1.
23. Zink, B., Diener, P., Pazos, E., and Tiglio, M. (2006). Cauchy-perturbative matching reexamined: Tests in spherical symmetry. Phys. Rev. D 73, 084011, gr-gc/0511163.
24. Goodale, T., Allen, G., Lanfermann, G., Massó, J., Radke, T., Seidel, E., and Shalf, J. (2003). The cactus framework and toolkit: Design and applications. In Vector and Parallel Processing - VECPAR'2002, 5th International Conference, Lecture Notes in Computer Science, Springer, Berlin. URL http://www.cactuscode.org/Publications/
25. Cactus Computational Toolkit home page, URL http://www.cactuscode.org/
26. Schnetter, E., Hawley, S. H., and Hawke, I. (2004). Evolutions in 3D numerical relativity using fixed mesh refinement Class. Quantum Grav. 21, 1465, qc/0310042.
27. Mesh Refinement with Carpet, URL http://www.carpetcode.org/.
28. Schnetter, E., Diener, P., Dorband, N., and Tiglio, M. (2006). A multi-block infrastructure for three-dimensional time-dependent numerical relativity. Class. Quantum Grav. 23, S 553 gr-qc/0602104.
29. LAPACK: Linear Algebra Package, URL http://www.netlib.org/lapack/.
30. BLAS: Basic Linear Algebra Subroutines, URL http://www.netlib.org/bl6as/.
31. Netlib Repository, URL http://www.netlib.org/.
32. Burns, G., Daoud, R., and Vaigl, J. (1994). LAM: An Open Cluster Environment for MPI. In Proceedings of Supercomputing Symposium, pp. 379-386, URL http://www.lam-mpi.org/download/files/lam-papers.tar.gz.
33. Squyres, J. M., and Lumsdaine, A. (2003). A Component Architecture for LAM/MPI. In Proceedings, 10th European PVM/MPI Users' Group Meeting (Springer-Verlag, Venice, Italy, 2003), no. 2840 in Lecture Notes in Computer Science, pp. 379-387.
34. LAM: LAM/MPI Parallel Computing, URL http://www.lam-mpi.org/
35. Gropp, W., Lusk, E., Doss, N., and Skjellum, A. (1996). A high-performance, portable implementation of the MPI message passing interface standard Parallel Computing 22, 789.
36. Gropp, W. D., and Lusk, E. (1996). User's Guide for mpich, a Portable Implementation of MPI, Mathematics and Computer Science Division, Argonne National Laboratory, ANL-96/6.
37. MPICH: ANL/MSU MPI implementation, URL http://www-unix.mcs.anl.gov/mpi/ mpich/.
38. MPI: Message Passing Interface Forum, URL http://www.mpi-forum.org/.
39. Sonderforschungsbereich/Transregio 7 "Gravitational Wave Astronomy", URL http://www.tpi.uni-jena.de/SFB/.
40. CCT Numerical Relativity, URL http://www.cct.lsu.edu/about/focus/ numerical/