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Hermite methods for hyperbolic initial-boundary value problems


Authors: John Goodrich, Thomas Hagstrom and Jens Lorenz
Journal: Math. Comp. 75 (2006), 595-630
MSC (2000): Primary 35G20
DOI: https://doi.org/10.1090/S0025-5718-05-01808-9
Published electronically: December 16, 2005
MathSciNet review: 2196982
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Abstract: We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using derivatives of order $ m$ in each coordinate direction are stable under $ m$-independent CFL constraints and converge at order $ 2m+1$. The stability proof relies on the fact that the Hermite interpolation process generally decreases a seminorm of the solution. We present numerical experiments demonstrating the resolution of the methods for large $ m$ as well as illustrating the basic theoretical results.


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Additional Information

John Goodrich
Affiliation: Acoustics Branch, NASA Glenn Research Center, Cleveland, Ohio 44135
Email: John.Goodrich@grc.nasa.gov

Thomas Hagstrom
Affiliation: Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, New Mexico 87131
Email: hagstrom@math.unm.edu

Jens Lorenz
Affiliation: Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, New Mexico 87131
Email: lorenz@math.unm.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01808-9
Keywords: High-order methods, hyperbolic problems, stability
Received by editor(s): January 7, 2004
Received by editor(s) in revised form: February 4, 2005
Published electronically: December 16, 2005
Additional Notes: The second author was supported in part by NSF Grants DMS-9971772, DMS-0306285, NASA Contract NAG3-2692, and the Institute for Computational Mechanics in Propulsion (ICOMP), NASA Glenn Research Center, Cleveland, OH. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of NSF or NASA
The third author was supported in part by DOE Grant DE-FG03-98ER25235 and the Institute for Computational Mechanics in Propulsion (ICOMP), NASA Glenn Research Center.
Article copyright: © Copyright 2005 American Mathematical Society

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