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Keywords:
$P_NP_M$ DG scheme; piecewise polynomial; projection; reconstruction; least square; local continuous space time Galerkin method; discontinuous Galerkin; advection equation; conservation law; von Neumann stability analysis; time discretization
$P_NP_M$ DG scheme; piecewise polynomial; projection; reconstruction; least square; local continuous space time Galerkin method; discontinuous Galerkin; advection equation; conservation law; von Neumann stability analysis; time discretization
Summary:
We give a proof of the existence of a solution of reconstruction operators used in the $P_NP_M$ DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the $P_NP_M$ DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several $P_NP_M$ DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency of the schemes.
References:
[1] Adams, R. A.: Sobolev Spaces. Pure and Applied Mathematics 65, Academic Press, New York (1975). DOI 10.1016/S0079-8169(08)61377-X | MR 0450957 | Zbl 0314.46030
[2] Badenjki, A.: The $P_NP_M$ DG Schemes for the One Dimensional Hyperbolic Conservation Laws. Doctoral Thesis, Otto-von-Guericke University, Magdeburg (2018).
[3] Cockburn, B.: An introduction to the discontinuous Galerkin method for convection-dominated problems. Advanced Numerical Approximation of Nonlinear Hyperbolic Equations A. Quarteroni et al. Lecture Notes in Mathematics 1697, Springer, Berlin (1998), 151-268. DOI 10.1007/BFb0096353 | MR 1728854 | Zbl 0927.65120
[4] Cockburn, B., Shu, C.-W.: TVB Runge Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework. Math. Comput. 52 (1989), 411-435. DOI 10.2307/2008474 | MR 0983311 | Zbl 0662.65083
[5] Dumbser, M., Balsara, D. S., Toro, E. F., Munz, C.-D.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227 (2008), 8209-8253. DOI 10.1016/j.jcp.2008.05.025 | MR 2446488 | Zbl 1147.65075
[6] Evans, L. C.: Partial Differential Equations. Graduate Studies in Mathematics 19, American Mathematical Society, Providence (1998). DOI 10.1090/gsm/019 | MR 1625845 | Zbl 1194.35001
[7] Goetz, C. R., Dumbser, M.: A square entropy stable flux limiter for $P_NP_M$ schemes. Available at https://arxiv.org/abs/1612.04793 (2016), 24 pages.
[8] Hirsch, C.: Numerical Computation of Internal and External Flows. Volume 1: Fundamentals of Numerical Discretization. Wiley Series in Numerical Methods in Engineering; Wiley, Chichester (1988). Zbl 0662.76001
[9] Koornwinder, T. H., Wong, R., Koekoek, R., Swarttouw, R. F.: Orthogonal polynomials. NIST Handbook of Mathematical Functions F. W. J. Olver et al. Cambridge University Press, Cambridge (2010), 435-484. MR 2655358 | Zbl 1198.00002
[10] Stegun, I. A.: Legendre functions. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables M. Abramowitz, I. A. Stegun National Bureau of Standards Applied Mathematics Series 55, Government Printing Office, Washington (1970). MR 0167642 | Zbl 0171.38503
[11] Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley (2003). MR 3058665 | Zbl 1046.15001
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