Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations
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Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.
KeywordsHigh-order nodal discontinuous Galerkin methods Maxwell equations Numerical dispersion and dissipation Strong-stability-preserving Runge-Kutta methods
- 5.Carpenter, M.H., Kennedy, C.A.: Fourth order 2N-storage Runge-Kutta scheme. NASA-TM-109112, NASA Langley Research Center, VA (1994) Google Scholar
- 11.Cockburn, B., Karniadakis, G.E., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods, Newport, RI, 1999. Lecture Notes in Comput. Sci. Eng., vol. 11, pp. 3–50. Springer, Berlin (2000) Google Scholar
- 16.Hesthaven, J.S.: High-order accurate methods in time-domain computational electromagnetics. A review. Adv. Imaging Electron Phys. 127(1), 59–123 (2003) Google Scholar
- 32.Sherwin, S.: Dispersion analysis of the continuous and discontinuous Galerkin formulations. In: Discontinuous Galerkin Methods, Newport, RI, 1999. Lecture Notes in Comput. Sci. Eng., vol. 11, pp. 425–431. Springer, Berlin (2000) Google Scholar