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An analysis of the discontinuous Galerkin method for wave propagation problems. (English) Zbl 0933.65113
Summary: The dispersion and dissipation properties of the discontinuous Galerkin method are investigated with a view to simulating wave propagation phenomena. These properties are analyzed in the semidiscrete context of the one-dimensional scalar advection equation and the two-dimensional wave equation, discretized on triangular and quadrilateral elements. They are verified by the results from full numerical solutions of the simple scalar advection equation and the Euler equations. $$\copyright$$ Academic Press.

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
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##### References:
 [1] Abboud, N.N.; Pinsky, P.M., Finite-element dispersion analysis for the 3-dimensional 2nd-order scalar wave-equation, Int. J. numer. methods eng., 35, 1183, (1992) · Zbl 0764.76033 [2] Atkins, H.L.; Shu, C.-W., Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, (1996) [3] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible navier – stokes equations, J. comput. phys., 131, 267, (1997) · Zbl 0871.76040 [4] Bey, K.S.; Oden, J.T., h-p, Comput. methods appl. mech. eng., 133, 259, (1996) [5] Biswas, R.; Devine, K.D.; Flaherty, J., Parallel, adaptive finite element methods for conservation laws, Appl. numer. math., 14, 255, (1994) · Zbl 0826.65084 [6] Bossavit, A., A rationale for “edge elements” in 3-D fields computations, IEEE trans. mag., 24, 74, (1988) [7] Chavent, G.; Cockburn, B., The local projectionP0P1, RAIRO, model. math. anal. numer., 23, 565, (1989) [8] Chaffin, D.J.; Baker, A.J., On Taylor weak statement finite-element methods for computational fluid dynamics, Int. J. numer. methods fluids, 21, 273, (1995) · Zbl 0840.76033 [9] Cockburn, B.; Shu, C.-W., The runge – kutta local projectionP1, RAIRO, model. math. anal. numer., 25, 337, (1991) [10] Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. II. general framework, Math. comput., 52, 411, (1989) · Zbl 0662.65083 [11] Cockburn, B.; Lin, S.Y.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. III. one dimensional systems, J. comput. phys., 84, 90, (1989) · Zbl 0677.65093 [12] Cockburn, B.; Hou, S.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. the multidimensional case, Math. comput., 54, 545, (1990) · Zbl 0695.65066 [13] Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin finite-element method for conservation laws. V. multidimensional systems, J. comput. phys., 141, 199, (1998) · Zbl 0920.65059 [14] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems, (1997) [15] Godunov, S.K., Finite difference method for numerical computation of discontinuous solutions of fluid dynamics, Math. sbornik., 47, 251, (1959) [16] Hu, F.Q.; Hussaini, M.Y.; Manthey, J.L., Application of low dissipation and dispersion runge – kutta schemes to benchmark problems in computational aeroacoustics, Proceedings, ICASE/larc workshop on benchmark problems in computational aeroacoustics, hampton, virginia, October 1994, (1995) · Zbl 0849.76046 [17] Hu, F.Q.; Hussaini, M.Y.; Manthey, J.L., Low-dissipation and low-dispersion runge – kutta schemes for computational acoustics, J. comput. phys., 124, 177, (1996) · Zbl 0849.76046 [18] Khelifa, A.; Ouellet, Y., Analysis of time-varying errors in quadratic finite-element approximation of hyperbolic problems, Int. J. numer. methods eng., 38, 3933, (1995) · Zbl 0854.65076 [19] Johnson, C.; Pitk\"aranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. comput., 46, 1, (1986) · Zbl 0618.65105 [20] Lele, S.K., Compact finite difference schemes with spectral-like resolution, J. comput. phys., 103, 16, (1992) · Zbl 0759.65006 [21] Lesaint, P.; Raviart, P.A., On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations, 89, (1974) [22] Liu, Y., Fourier analysis of numerical algorithms for the Maxwell equations, J. comput. phys., 124, 396, (1996) · Zbl 0858.65125 [23] Lowrie, R.B., Compact higher-order numerical methods for hyperbolic conservation laws, (1996) [24] Lowrie, R.B.; Roe, P.L.; Van Leer, B., A space-time discontinuous Galerkin method for the time-accurate numerical solution of hyperbolic conservation laws, (1995) [25] Makridakis, C.; Monk, P., Time-discrete finite element schemes for Maxwell’s equations, RAIRO, model. math. anal. numer., 29, 171, (1995) · Zbl 0834.65120 [26] Mur, G., The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media, IEEE trans. mag., 28, 1130, (1992) [27] Peterson, T., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. numer. anal., 28, 133, (1991) · Zbl 0729.65085 [28] Reed, W.H.; Hill, T.R., Triangular mesh methods for the neutron transport equation, (1973) [29] Richter, G.R., An optimal-order error estimate for the discontinuous Galerkin method, Math. comput., 50, 75, (1988) · Zbl 0643.65059 [30] Shakib, F.; Hughes, T.J.R., A new finite element formulation for computational fluid dynamics. IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. methods appl. mech. eng., 87, 35, (1991) · Zbl 0760.76051 [31] Shang, J.S., High-order compact-difference schemes for time-dependent Maxwell equations, (1998) · Zbl 0956.78018 [32] Steger, J.; Warming, R.F., Flux vector splitting of the inviscid gas-dynamic equations with applications to the finite difference methods, J. comput. phys., 40, 263, (1981) · Zbl 0468.76066 [33] Tam, C.K.W.; Webb, J.W., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. comput. phys., 107, 262, (1993) · Zbl 0790.76057 [34] Thompson, L.L.; Pinsky, P.M., Complex wave-number Fourier-analysis of the p-version finite element method, Comput. mech., 13, 255, (1994) · Zbl 0789.73076 [35] Waburton, T.C.; Lomtev, I.; Kirby, R.M.; Karniadakis, G., A discontinuous Galerkin method for the compressible navier – stokes equations on hybrid grids, (1997) [36] Wu, J.Y.; Lee, R., The advantages of triangular and tetrahedral edge elements for electromagnectic modeling with the finite-element method, IEEE trans. antennas and propagation, 45, 1431, (1997)
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