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P-adaptive Hermite methods for initial value problems∗

Published online by Cambridge University Press:  11 January 2012

Ronald Chen  and
Thomas Hagstrom
Ronald Chen
Affiliation:
Department of Mathematics and Statistics, The University of New Mexico, MSC03 2150, Albuquerque, 87131-0001 NM, USA. xqroy@math.unm.edu
Thomas Hagstrom
Affiliation:
Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, 75275-0156 TX, USA; thagstrom@smu.edu
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Abstract

We study order-adaptive implementations of Hermite methods for hyperbolic and singularly perturbed parabolic initial value problems. Exploiting the facts that Hermite methods allow the degree of the local polynomial representation to vary arbitrarily from cell to cell and that, for hyperbolic problems, each cell can be evolved independently over a time-step determined only by the cell size, a relatively straightforward method is proposed. Its utility is demonstrated on a number of model problems posed in 1+1 and 2+1 dimensions.

Keywords

Adaptivityhigh-order methods
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 3: Special volume in honor of Professor David Gottlieb , May 2012 , pp. 545 - 557
Copyright
© EDP Sciences, SMAI, 2012

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References

Ainsworth, M., Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42 (2004) 553–575. Google Scholar
Ainsworth, M., Dispersive and dissipative behavior of high-order discontinuous Galerkin finite element methods. J. Comput. Phys. 198 (2004) 106–130. Google Scholar
D. Appelö and T. Hagstrom, Experiments with Hermite methods for simulating compressible flows : Runge-Kutta time-stepping and absorbing layers, in 13th AIAA/CEAS Aeroacoustics Conference. AIAA (2007).
Birkhoff, G., Schultz, M. and Varga, R., Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11 (1968) 232–256. Google Scholar
P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities. Springer-Verlag, New York (1995).
P. Davis, Interpolation and Approximation. Dover Publications, New York (1975).
L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz and A. Zdunek, Computing with hp -Adaptive Finite Elements. Applied Mathematics & Nonlinear Science, Chapman & Hall/CRC, Boca Raton (2007).
C. Dodson, A high-order Hermite compressible Navier-Stokes solver. Master’s thesis, The University of New Mexico (2003).
Fornberg, B., On a Fourier method for the integration of hyperbolic equations. SIAM J. Numer. Anal. 12 (1975) 509–528. Google Scholar
Goodrich, J., Hagstrom, T. and Lorenz, J., Hermite methods for hyperbolic initial-boundary value problems. Math. Comput. 75 (2006) 595–630. Google Scholar
D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977).
Gottlieb, D. and Tadmor, E., The CFL condition for spectral approximations to hyperbolic initial-boundary value problems. Math. Comput. 56 (1991) 565–588. Google Scholar
A. Griewank, Evaluating Derivatives : Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2000).
Hairer, E., Lubich, C. and Schlichte, M., Fast numerical solution of nonlinear Volterra convolutional equations. SIAM J. Sci. Statist. Comput. 6 (1985) 532–541. Google Scholar
Jiang, G.-S. and Tadmor, E., Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892–1917. Google Scholar
Kreiss, H.-O. and Oliger, J., Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24 (1972) 199–215. Google Scholar
Lörcher, F., Gassner, G. and Munz, C.-D., An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227 (2008) 5649–5670. Google Scholar
Warburton, T. and Hagstrom, T., Taming the CFL number for discontinuous Galerkin methods on structured meshes. SIAM J. Numer. Anal. 46 (2008) 3151–3180. Google Scholar
Weideman, J. and Trefethen, L., The eigenvalues of second-order differentiation matrices. SIAM J. Numer. Anal. 25 (1988) 1279–1298. Google Scholar