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An analysis of the spectrum of the discontinuous Galerkin method. (English) Zbl 1255.65166
Summary: We derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal \([p/p+1]\) Padé approximation of \(\exp(-z)\) when pth degree basis functions are used. We derive an upper bound on the eigenvalue with the largest magnitude as \((p+1)(p+2)\). We demonstrate that this bound is not tight and prove that the asymptotic growth rate of the spectral radius is slower than quadratic in \(p\). We also analyze the behavior of the spectrum near the imaginary axis to demonstrate that the spectral curves approach the imaginary axis although there are no purely imaginary eigenvalues.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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