Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Journal of Computational Physics
Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
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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] Pade 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.