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Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics. (English) Zbl 1063.65113
Summary: A new Runge-Kutta method with minimal dispersion and dissipation error is developed. The Chebyshev pseudospectral method is utilized using spatial discretization and a new fourth-order six-stage Runge-Kutta scheme is used for time advancing. The proposed scheme is more efficient than the existing ones for acoustic computations.

##### MSC:
 65M70 Spectral, collocation and related methods (IVP of PDE) 35L05 Wave equation (hyperbolic PDE) 76Q05 Hydro- and aero-acoustics 76M22 Spectral methods (fluid mechanics) 65L06 Multistep, Runge-Kutta, and extrapolation methods 65M15 Error bounds (IVP of PDE)
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##### References:
 [1] Albrecht, P.: The Runge -- Kutta theory in a nutshell. SIAM J. Numer. anal. 33, No. 5, 1712-1735 (1996) · Zbl 0858.65074 [2] Bogey, C.; Bailly, C.: A family of low dispersive and low dissipative explicit schemes for flow and noise computations. J. comput. Phys. 194, 194-214 (2004) · Zbl 1042.76044 [3] Butcher, J. C.: The numerical analysis of ordinary differential equationsrunge -- Kutta and general linear methods. (1987) · Zbl 0616.65072 [4] 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 [5] Kosloff, D.; Tal-Ezer, J.: A modified Chebyshev pseudospectral method with $O(N-1)$ time step restriction. J. comput. Phys. 104, 457-469 (1993) · Zbl 0781.65082 [6] Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters. SIAM J. Appl. math. 11, 431-441 (1963) · Zbl 0112.10505 [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] 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 [9] Tam, C. K. W.: Computational aeroacousticsissues and methods. Aiaa j. 33, 1788-1796 (1995) · Zbl 0856.76080 [10] Van Der Howen, P. J.; Sommeijer, B. P.: Explicit Runge -- Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions. SIAM J. Numer. anal. 24, 595-617 (1987) · Zbl 0624.65058