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Fast high order ADER schemes for linear hyperbolic equations. (English) Zbl 1052.65078
Summary: A reformulation of the ADER approach (Arbitrary high order schemes using DERivatives) for linear hyperbolic partial differential equations is presented. This reformulation leads to a drastic decrease of the computational effort. A formula for the construction of ADER schemes that are arbitrary high order accurate in space and time is given. The accuracy for some selected schemes is shown numerically for the two-dimensional linearized Euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics.

65M06Finite difference methods (IVP of PDE)
76Q05Hydro- and aero-acoustics
76M12Finite volume methods (fluid mechanics)
65M12Stability and convergence of numerical methods (IVP of PDE)
35L45First order hyperbolic systems, initial value problems
65M15Error bounds (IVP of PDE)
Full Text: DOI
[1] Ben-Artzi, M.; Falcovitz, J.: A second-order Godunov-type scheme for compressible fluid dynamics. J. comput. Phys. 55, 1-32 (1984) · Zbl 0535.76070
[2] Butcher, J. C.: The numerical analysis of ordinary differential equations: Runge--Kutta and general linear methods. (1987) · Zbl 0616.65072
[3] Casper, J.; Atkins, H. L.: A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. J. comput. Phys. 106, 62-76 (1993) · Zbl 0774.65066
[4] Cockburn, B.; Johnson, C.; Shu, C. -W; Tadmor, E.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Lecture notes in mathematics 1697, 325-432 (1998)
[5] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R.: Uniformly high order accurate essentially non-oscillatory schemes III. J. comput. Phys. 71, 231-303 (1987) · Zbl 0652.65067
[6] Hirsch, C.: Numerical computation of internal and external flows vol II: Computational methods for inviscid and viscous flow. (1988)
[7] T. Schwartzkopff, C.-D. Munz, E.F. Toro, R.C. Millington, ADER-2d: A very high-order approach for linear hyperbolic systems. In: Proceedings of ECCOMAS CFD Conference 2001, September 2001, ISBN 0 905 091 12 4 · Zbl 1028.76030
[8] Schwartzkopff, T.; Munz, C. -D; Toro, E. F.; Millington, R. C.: The ADER approach in 2d.. Discrete modelling and discrete algorithms on continuum mechanics, 207-216 (2001) · Zbl 1028.76030
[9] Schwartzkopff, T.; Munz, C. -D; Toro, E. F.: ADER: A high order approach for linear hyperbolic systems in 2d. J. sci. Comput. 17, No. 1-4, 231-240 (2002) · Zbl 1022.76034
[10] C.K.W. Tam, Numerical methods in computational aeroacoustics. Von Karman Institute for Fluid Dynamics, Lecture Series 1996-04, Applied Aero-Acoutistics: Prediction Methods, 1996
[11] Toro, E. F.; Millington, R. C.; Nejad, L. A. M: Towards very high-order Godunov schemes. Godunov methods: theory and applications, 905-937 (2001) · Zbl 0989.65094
[12] Toro, E. F.: Riemann solvers and numerical methods for fluid dynamics. (1997) · Zbl 0888.76001
[13] E.F. Toro and R.C. Millington, ADER: High-order non-oscillatory advection schemes. In: Proceedings of the 8th International Conference on Nonlinear Hyperbolic Problems, February 2000 (preprint)