Optimal Runge-Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems. (English) Zbl 1242.65190
Summary: We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge-Kutta time integrators, with the aim of deriving optimal Runge-Kutta schemes for wave propagation applications. We review relevant Runge-Kutta methods from literature, and consider schemes of order \(q\) from 3 to 4, and number of stages up to \(q + 4\), for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined.
In the first one, the element size is totally free, and a 8-stage, fourth-order Runge-Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge-Kutta methods, we provide the coefficients for a \(2N\)-storage implementation, along with the information needed by the user to employ them optimally.
In the first one, the element size is totally free, and a 8-stage, fourth-order Runge-Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge-Kutta methods, we provide the coefficients for a \(2N\)-storage implementation, along with the information needed by the user to employ them optimally.
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 35L05 | Wave equation |
| 65Y20 | Complexity and performance of numerical algorithms |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
discontinuous Galerkin method; Runge-Kutta method; wave propagation; computational efficiency; wave equation; semidiscretization; numerical examples; stabilitySoftware:
OctaveReferences:
| [1] | Colonius, T.; Lele, S. K., Computational aeroacoustics: progress on nonlinear problems of sound generation, Prog. Aerosp. Sci., 40, 345-416 (2004) |
| [2] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261 (2001) · Zbl 1065.76135 |
| [3] | Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, vol. 54 (2008), Springer-Verlag: Springer-Verlag New York, USA · Zbl 1134.65068 |
| [4] | Williamson, J. H., Low-storage Runge-Kutta schemes, J. Comput. Phys., 35, 48-56 (1980) · Zbl 0425.65038 |
| [5] | Calvo, M.; Franco, J. M.; Rández, L., Minimum storage Runge-Kutta schemes for computational acoustics, Comput. Math. Appl., 45, 535-545 (2003) · Zbl 1036.65059 |
| [6] | M.H. Carpenter, C.A. Kennedy, Fourth-Order \(2N\); M.H. Carpenter, C.A. Kennedy, Fourth-Order \(2N\) |
| [7] | Mead, J. L.; Renaut, R. A., Optimal Runge-Kutta methods for first order pseudospectral operators, J. Comput. Phys., 152, 404-419 (1999) · Zbl 0935.65100 |
| [8] | Allampalli, V.; Hixon, R.; Nallasamy, M.; Sawyer, S. D., High-accuracy large-step explicit Runge-Kutta (HALE-RK) schemes for computational aeroacoustics, J. Comput. Phys., 228, 3837-3850 (2009) · Zbl 1171.76039 |
| [9] | Hu, F. Q.; Hussaini, M. Y.; Manthey, J. L., Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics, J. Comput. Phys., 124, 177-191 (1996) · Zbl 0849.76046 |
| [10] | Stanescu, D.; Habashi, W. G., \(2N\)-storage low dissipation and dispersion Runge-Kutta schemes for computational acoustics, J. Comput. Phys., 143, 674-681 (1998) · Zbl 0952.76063 |
| [11] | Berland, J.; Bogey, C.; Bailly, C., Low-dissipation and low-dispersion fourth-order Runge-Kutta algorithm, Comput. Fluids, 35, 1459-1463 (2006) · Zbl 1177.76252 |
| [12] | Calvo, M.; Franco, J. M.; Rández, L., A new minimum storage Runge-Kutta scheme for computational acoustics, J. Comput. Phys., 201, 1-12 (2004) · Zbl 1059.65061 |
| [13] | Tselios, K.; Simos, T., Optimized Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics, Phys. Lett. A, 363, 38-47 (2007) · Zbl 1197.76082 |
| [14] | Pirozzoli, S., Performance analysis and optimization of finite-difference schemes for wave propagation problems, J. Comput. Phys., 222, 809-831 (2007) · Zbl 1158.76382 |
| [15] | Bernardini, M.; Pirozzoli, S., A general strategy for the optimization of Runge-Kutta schemes for wave propagation phenomena, J. Comput. Phys., 228, 4182-4199 (2009) · Zbl 1273.76294 |
| [16] | Toulorge, T.; Desmet, W., CFL conditions for Runge-Kutta discontinuous Galerkin methods on triangular grids, J. Comput. Phys., 230, 4657-4678 (2011) · Zbl 1220.65139 |
| [17] | Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations (2003), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester, England · Zbl 1032.65512 |
| [18] | Baldauf, M., Stability analysis for linear discretisations of the advection equation with Runge-Kutta time integration, J. Comput. Phys., 227, 6638-6659 (2008) · Zbl 1149.65064 |
| [19] | Ramboer, J.; Broeckhoven, T.; Smirnov, S.; Lacor, C., Optimization of time integration schemes coupled to spatial discretization for use in caa applications, J. Comput. Phys., 213, 777-802 (2006) · Zbl 1136.76391 |
| [20] | Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 89-112 (2001) · Zbl 0967.65098 |
| [21] | Ruuth, S. J., Global optimization of explicit strong-stability-preserving Runge-Kutta methods, Math. Comput., 75, 183-207 (2006) · Zbl 1080.65088 |
| [22] | Jameson, A., Solution of the Euler equations for two dimensional transonic flow by a multigrid method, Appl. Math. Comput., 13, 327-355 (1983) · Zbl 0545.76065 |
| [23] | Denia, F. D.; Albelda, J.; Fuenmayor, F. J.; Torregrosa, A. J., Acoustic behaviour of elliptical chamber mufflers, J. Sound Vib., 241, 401-421 (2001) |
| [24] | Hong, K.; Kim, J., Natural mode analysis of hollow and annular elliptical cylindrical cavities, J. Sound Vib., 183, 327-351 (1995) · Zbl 0982.76550 |
| [25] | Toulorge, T.; Desmet, W., Curved boundary treatments for the discontinuous Galerkin method applied to aeroacoustic propagation, AIAA J., 48, 479-489 (2010) |
| [26] | J.W. Eaton, D. Bateman, S. Hauberg, GNU Octave Manual Version 3, Network Theory Limited, Bristol, GB, 2008.; J.W. Eaton, D. Bateman, S. Hauberg, GNU Octave Manual Version 3, Network Theory Limited, Bristol, GB, 2008. |
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