Discontinuous Galerkin spectral element approximations on moving meshes. (English) Zbl 1210.65164
Summary: We derive and evaluate high order space arbitrary Lagrangian-Eulerian (ALE) methods to compute conservation laws on moving meshes to the same time order as on a static mesh. We use a discontinuous Galerkin spectral element method (DGSEM) in space, and one of a family of explicit time integrators such as Adams-Bashforth or low storage explicit Runge-Kutta. The approximations preserve the discrete metric identities and the discrete geometric conservation law (DGCL) by construction. We present time-step refinement studies with moving meshes to validate the approximations.
The test problems include propagation of an electromagnetic Gaussian plane wave, a cylindrical pressure wave propagating in a subsonic flow, and a vortex convecting in a uniform inviscid subsonic flow. Each problem is computed on a time-deforming mesh with three methods used to calculate the mesh velocities: from exact differentiation, from the integration of an acceleration equation, and from numerical differentiation of the mesh position.
The test problems include propagation of an electromagnetic Gaussian plane wave, a cylindrical pressure wave propagating in a subsonic flow, and a vortex convecting in a uniform inviscid subsonic flow. Each problem is computed on a time-deforming mesh with three methods used to calculate the mesh velocities: from exact differentiation, from the integration of an acceleration equation, and from numerical differentiation of the mesh position.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 76G25 | General aerodynamics and subsonic flows |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 78M10 | Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory |
Keywords:
discontinuous Galerkin spectral element method; moving mesh; arbitrary Lagrangian-Eulerian method; discrete geometric conservation law; numerical examples; electromagnetic Gaussian plane wave; cylindrical pressure wave; numerical differentiationSoftware:
HE-E1GODFReferences:
| [1] | Ainsworth, M., Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods, Journal of Computational Physics, 198, 106-130 (2004) · Zbl 1058.65103 |
| [2] | Zuijlen, A.; Bijl, A., High-order time integration through smooth mesh deformation for 3d fluid-structure interaction simulations, Journal of Computational Physics, 224, 414-430 (2007) · Zbl 1261.74012 |
| [3] | Bjøntegaard, T.; Rønquist, Einar M., Accurate interface-tracking for arbitrary Lagrangian-Eulerian schemes, Journal of Computational Physics, 228, 4379-4399 (2009) · Zbl 1395.76055 |
| [4] | Black, K., A conservative spectral element method for the approximation of compressible fluid flow, Kybernetika, 35, 1, 133-146 (1999) · Zbl 1274.76271 |
| [5] | Piquette, J. C.; Van Buren, A. L.; Rogers, P. H., Censor’s acoustical Doppler effect analysis – is it a valid method?, Journal of the Acoustical Society of America, 83, 1681-1682 (1988) |
| [6] | Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2003), John Wiley · Zbl 1032.65512 |
| [7] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods: Fundamentals in Single Domains (2006), Springer · Zbl 1093.76002 |
| [8] | M.H. Carpenter, C.A. Kennedy, Fourth-order 2N-storage Runge-Kutta schemes, NASA Technical Memorandum 109112, 1994.; M.H. Carpenter, C.A. Kennedy, Fourth-order 2N-storage Runge-Kutta schemes, NASA Technical Memorandum 109112, 1994. |
| [9] | Censor, D., Acoustical Doppler effect analysis – is it a valid method?, Journal of the Acoustical Society of America, 83, 1223-1230 (1988) |
| [10] | Censor, D., Non-relativistic scattering by time-varying bodies and media, Progress in Electromagnetics Research, PIER, 48, 249-278 (2004) |
| [11] | Koobus, B.; Farhat, C., Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes, Computer Methods in Applied Mechanics and Engineering, 170, 103-129 (1999) · Zbl 0943.76055 |
| [12] | Visbal, M.; Gaitonde, D., On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, Journal of Computational Physics, 181, 155-185 (2002) · Zbl 1008.65062 |
| [13] | Etienne, S.; Garon, A.; Pelletier, D., Perspective on the geometric conservation law and finite element methods for ale simulations of incompressible flow, Journal of Computational Physics, 228, 2313-2333 (2009) · Zbl 1275.76159 |
| [14] | Hesthaven, J. S., High-order Accurate Methods in Time-domain Computational Electromagnetics: A Review, Advances in Imaging and Electron Physics, vol. 127 (2003), Academic Press, pp. 59-123 |
| [15] | Ho, M., Scattering of electromagnetic waves from vibrating perfect surfaces: simulation using relativistic boundary conditions, Journal of Electromagnetic Waves and Applications, 20, 4, 425-433 (2006) |
| [16] | Hu, Fang Q.; Hussaini, M. Y.; Rasetarinera, Patrick, An analysis of the discontinuous Galerkin method for wave propagation problems, Journal of Computational Physics, 151, 2, 921-946 (1999) · Zbl 0933.65113 |
| [17] | Hu, F. Q.; Hussaini, M. Y.; Manthey, J. L., Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, Journal of Computational Physics, 124, 177-191 (1996) · Zbl 0849.76046 |
| [18] | Lomtev, I.; Kirby, R.; Karniadakis, E., A discontinuous Galerkin ALE method for compressible viscous flows in moving domains, Journal of Computational Physics, 155, 128-159 (1999) · Zbl 0956.76046 |
| [19] | Kopriva, D., Metric identities and the discontinuous spectral element method on curvilinear meshes, Journal of Scientific Computing, 26, 3, 301-327 (2006) · Zbl 1178.76269 |
| [20] | Kopriva, D., Implementing Spectral Methods for Partial Differential Equations (2009), Springer: Springer Netherlands · Zbl 1172.65001 |
| [21] | Mavriplis, D. J.; Yang, Z., Construction of the discrete geometric conservation law for high-order time-accurate simulation on dynamic meshes, Journal of Computational Physics, 213, 557-573 (2006) · Zbl 1136.76402 |
| [22] | Morse, P.; Uno Ingard, K., Theoretical Acoustics (1986), Princeton University Press |
| [23] | Wunenburger, R.; Mujica, N.; Fauve, S., Experimental study of the Doppler shift generated by a vibrating scatterer, Journal of the Acoustical Society of America, 115, 507-514 (2004) |
| [24] | Ostashev, V., Acoustics in Moving Inhomogeneous Media (1997), E & FN SPON · Zbl 0896.76085 |
| [25] | K. Ou, CH. Liang, A. Jameson, A high-order spectral difference method for the Navier-Stokes equations on unstructured moving deformable grids, in: 48th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, January 2010.; K. Ou, CH. Liang, A. Jameson, A high-order spectral difference method for the Navier-Stokes equations on unstructured moving deformable grids, in: 48th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, January 2010. |
| [26] | Shang, J. S., Characteristic-based Methods in Computational Electromagnetics. Characteristic-based Methods in Computational Electromagnetics, Computational Electromagnetics and Its Applications (1997), Kluwer Academic Publishers, pp. 189-211 |
| [27] | Shiozawa, T., Classical Relativistic Electrodynamics: Theory of Light Emission and Application to Free Electron Lasers (2004), Springer-Verlag: Springer-Verlag Berlin, Heidelberg |
| [28] | Harfoush, F. A.; Taflove, A.; Kriegsmann, G. A., A numerical technique for analyzing electromagnetic wave scattering from moving surfaces, IEEE Transactions on Antennas and Propagation, 37, 1, 55-63 (1989) |
| [29] | Tam, C. K.W., Computational aeroacoustics: an overview of computational challenges and applications, International Journal of Computational Fluid Dynamics, 18, 547-567 (2004) · Zbl 1065.76180 |
| [30] | Thompson, J. F.; Soni, B. K.; Weatherill, N. P., Handbook of grid generation (1999), CRC Press · Zbl 0980.65500 |
| [31] | Thompson, J. F.; Warsi, Z. U.A.; Mastin, C. W., Boundary-fitted coordinate systems for numerical solution of partial-differential equations – a review, Journal of Computational Physics, 47, 1, 1-108 (1982) · Zbl 0492.65011 |
| [32] | Toro, Eleuterio F., Riemann Solvers and Numerical Methods for Fluid Dynamics - A Practical Introduction (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0888.76001 |
| [33] | Vinokur, M., Conservation equations of gas-dynamics in curvilinear coordinate systems, Journal of Computational Physics, 14, 105-125 (1974) · Zbl 0277.76061 |
| [34] | Williamson, J. H., Low storage Runge-Kutta schemes, Journal of Computational Physics, 35, 48-56 (1980) · Zbl 0425.65038 |
| [35] | Kopriva, D.; Woodruff, S.; Hussaini, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, International Journal of Numerical Methods and Engineering, 53, 105-122 (2002) · Zbl 0994.78020 |
| [36] | Mujica, N.; Wunenburger, R.; Fauve, S., Scattering of a sound wave by a vibrating surface, The European Physical Journal B, 33, 209-213 (2003) |
| [37] | Lu, T.; Zhang, P.; Cai, W., Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions, Journal of Computational Physics, 200, 2, 549-580 (2004) · Zbl 1115.78330 |
| [38] | De Zutter, D., Reflections from linearly vibrating objects: plane mirror at oblique incidence, IEEE Transactions on Antennas and Propagation, 5, 898-901 (1982) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
