P-adaptive Hermite methods for initial value problems
ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 545-557.

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.

DOI : 10.1051/m2an/2011050
Classification : 65M70, 65M12
Mots-clés : adaptivity, high-order methods 
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     author = {Chen, Ronald and Hagstrom, Thomas},
     title = {$P$-adaptive {Hermite} methods for initial value problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {545--557},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {3},
     year = {2012},
     doi = {10.1051/m2an/2011050},
     zbl = {1272.65077},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2011050/}
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Chen, Ronald; Hagstrom, Thomas. $P$-adaptive Hermite methods for initial value problems. ESAIM: Mathematical Modelling and Numerical Analysis , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 545-557. doi : 10.1051/m2an/2011050. http://www.numdam.org/articles/10.1051/m2an/2011050/

[1] M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42 (2004) 553-575. | MR | Zbl

[2] M. Ainsworth, Dispersive and dissipative behavior of high-order discontinuous Galerkin finite element methods. J. Comput. Phys. 198 (2004) 106-130. | MR | Zbl

[3] 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).

[4] G. Birkhoff, M. Schultz and R. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations. Numer. Math. 11 (1968) 232-256. | MR | Zbl

[5] P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities. Springer-Verlag, New York (1995). | MR | Zbl

[6] P. Davis, Interpolation and Approximation. Dover Publications, New York (1975). | MR | Zbl

[7] 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). | Zbl

[8] C. Dodson, A high-order Hermite compressible Navier-Stokes solver. Master's thesis, The University of New Mexico (2003).

[9] B. Fornberg, On a Fourier method for the integration of hyperbolic equations. SIAM J. Numer. Anal. 12 (1975) 509-528. | MR | Zbl

[10] J. Goodrich, T. Hagstrom and J. Lorenz, Hermite methods for hyperbolic initial-boundary value problems. Math. Comput. 75 (2006) 595-630. | MR | Zbl

[11] D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977). | Zbl

[12] D. Gottlieb and E. Tadmor, The CFL condition for spectral approximations to hyperbolic initial-boundary value problems. Math. Comput. 56 (1991) 565-588. | MR | Zbl

[13] A. Griewank, Evaluating Derivatives : Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2000). | MR | Zbl

[14] E. Hairer, C. Lubich and M. Schlichte, Fast numerical solution of nonlinear Volterra convolutional equations. SIAM J. Sci. Statist. Comput. 6 (1985) 532-541. | MR | Zbl

[15] G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. | MR | Zbl

[16] H.-O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24 (1972) 199-215. | MR

[17] F. Lörcher, G. Gassner and C.-D. Munz, An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227 (2008) 5649-5670. | MR | Zbl

[18] T. Warburton and T. Hagstrom, Taming the CFL number for discontinuous Galerkin methods on structured meshes. SIAM J. Numer. Anal. 46 (2008) 3151-3180. | MR | Zbl

[19] J. Weideman and L. Trefethen, The eigenvalues of second-order differentiation matrices. SIAM J. Numer. Anal. 25 (1988) 1279-1298. | MR | Zbl

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