P-adaptive Hermite methods for initial value problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, p. 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 : https://doi.org/10.1051/m2an/2011050
Classification:  65M70,  65M12
Keywords: adaptivity, high-order methods
@article{M2AN_2012__46_3_545_0,
     author = {Chen, Ronald and Hagstrom, Thomas},
     title = {$P$-adaptive Hermite methods for initial value problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {3},
     year = {2012},
     pages = {545-557},
     doi = {10.1051/m2an/2011050},
     zbl = {1272.65077},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_3_545_0}
}
Chen, Ronald; Hagstrom, Thomas. $P$-adaptive Hermite methods for initial value problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 545-557. doi : 10.1051/m2an/2011050. http://www.numdam.org/item/M2AN_2012__46_3_545_0/

[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 2084226 | Zbl 1074.65112

[2] M. Ainsworth, Dispersive and dissipative behavior of high-order discontinuous Galerkin finite element methods. J. Comput. Phys. 198 (2004) 106-130. | MR 2071391 | Zbl 1058.65103

[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 226817 | Zbl 0159.20904

[5] P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities. Springer-Verlag, New York (1995). | MR 1367960 | Zbl 0840.26002

[6] P. Davis, Interpolation and Approximation. Dover Publications, New York (1975). | MR 380189 | Zbl 0329.41010

[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 1148.65001

[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 421096 | Zbl 0349.35003

[10] J. Goodrich, T. Hagstrom and J. Lorenz, Hermite methods for hyperbolic initial-boundary value problems. Math. Comput. 75 (2006) 595-630. | MR 2196982 | Zbl 1103.35065

[11] D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods. SIAM, Philadelphia (1977). | Zbl 0412.65058

[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 1066833 | Zbl 0723.65079

[13] A. Griewank, Evaluating Derivatives : Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia (2000). | MR 1753583 | Zbl 1159.65026

[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 791183 | Zbl 0581.65095

[15] G.-S. Jiang and E. Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput. 19 (1998) 1892-1917. | MR 1638064 | Zbl 0914.65095

[16] H.-O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24 (1972) 199-215. | MR 319382

[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 2414924 | Zbl 1147.65077

[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 2439506 | Zbl 1181.35010

[19] J. Weideman and L. Trefethen, The eigenvalues of second-order differentiation matrices. SIAM J. Numer. Anal. 25 (1988) 1279-1298. | MR 972454 | Zbl 0666.65063