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An analysis of the spectrum of the discontinuous Galerkin method. II: Nonuniform grids. (English) Zbl 1282.65119
Summary: We analyze the eigenvalues of the discontinuous Galerkin spatial operator for the one-dimensional linear advection equation on nonuniform grids. We show that when the difference in cell sizes is below an order-dependent critical number, the spectrum continuously changes with changing cell sizes. In a particularly simple case where the mesh contains cells of only two sizes, the spectrum grows linearly with the proportion of smaller cells in the mesh. When the cell size scale is larger than a critical value, the eigenvalues corresponding to smaller cells separate from the rest of the spectrum. We provide an easily computable estimate of the upper bound on the spectrum for both cases. This estimate can be used to compute a relaxed time step restriction if smaller cells are dispersed among larger cells. Numerical examples for one- and two-dimensional problems reveal that computational saving can be realized by use of a larger stable time step.
For Part I by the authors see [ibid. 64, 1–18 (2013; Zbl 1255.65166)].

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Software:
Eigtool
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References:
[1] (Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, (1965), Dover New York) · Zbl 0171.38503
[2] Baker, G. A.; Graves-Morris, P. R., PadĂ© approximants, (1981), Addison-Wesley Reading, MA/Don Mills, ON · Zbl 0468.30033
[3] Bayyuk, S. A.; Powell, K. G.; van Leer, B., A simulation technique for 2-d unsteady inviscid flows around arbitrarily moving and deforming bodies of arbitrary geometry, (1995), AIAA-93-3391-CP
[4] Chavent, G.; Cockburn, B., The local projection \(P^0 P^1\) discontinuous Galerkin method for scalar conservation laws, RAIRO, Model. Math. Anal. Numer., 23, 565, (1989) · Zbl 0715.65079
[5] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin finite element method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[6] Crossley, A. J.; Wright, N. G., Time accurate local timestepping for the unsteady shallow water equations, Internat. J. Numer. Methods Fluids, 48, 775-779, (2005) · Zbl 1071.76033
[7] Godel, N.; Schomann, S.; Warburton, T.; Clemens, M., GPU accelerated Adams-bashforth multirate discontinuous Galerkin simulation of high frequency electromagnetic fields, IEEE Trans. Magn., 46, 8, 2735-2738, (2010)
[8] Hairer, E.; Norsett, S. P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, (2000), Springer Berlin
[9] Krivodonova, L., An efficient local time-stepping scheme for solution of nonlinear conservation laws, J. Comput. Phys., 229, 8537-8551, (2010) · Zbl 1201.65171
[10] Krivodonova, L.; Qin, R., An analysis of the spectrum of the discontinuous Galerkin method, Appl. Numer. Math., 64, 1-18, (2013) · Zbl 1255.65166
[11] Qin, R.; Krivodonova, L., A discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries, J. Comput. Sci., 4, 24-35, (2013)
[12] Trefethen, L. N.; Embree, M., Spectra and pseudospectra - the behavior of nonnormal matrices and operators, (2005), Princeton University Press Princeton, Oxford · Zbl 1085.15009
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