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Discretely nonreflecting boundary conditions for linear hyperbolic systems. (English) Zbl 0962.76066

Summary: Many compressible flow and aeroacoustic computations rely on accurate nonreflecting or radiation boundary conditions. When the equations and boundary conditions are discretized using a finite difference scheme, the dispersive nature of the discretized equations can lead to spurious numerical reflections not seen in the continuous boundary value problem. Here we construct discretely nonreflecting boundary conditions, which account for the particular finite-difference scheme used, and are designed to minimize these spurious numerical reflections. We obtain stable boundary conditions that are local and nonreflecting to arbitrarily high-order of accuracy, and present test cases for linearized Euler equations. For the cases presented, reflections for a pressure pulse leaving the boundary are reduced by up to two orders of magnitude over typical ad hoc closures, and for a vorticity pulse, reflections are reduced by up to four orders of magnitude.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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[1] Bayliss, A.; Turkel, E., Far field boundary conditions for compressible flows, J. Comput. Phys., 48, 182 (1982) · Zbl 0494.76072
[2] Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to high-order compact schemes, J. Comput. Phys., 111, 220 (1994) · Zbl 0832.65098
[3] Colonius, T., Numerically nonreflecting boundary and interface conditions for compressible flow and aeroacoustic computations, AIAA J., 35, 1126 (1997) · Zbl 0909.76058
[4] Colonius, T.; Lele, S. K.; Moin, P., Boundary conditions for direct computation of aerodynamic sound generation, AIAA J., 31, 1574 (1993) · Zbl 0785.76069
[5] Engquist, B.; Majda, A., Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31, 629 (1977) · Zbl 0367.65051
[6] Engquist, B.; Majda, A., Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math., 32, 313 (1979) · Zbl 0387.76070
[7] Giles, M. B., Nonreflecting boundary-conditions for Euler equation calculations, AIAA J., 28, 2050 (1990)
[8] Goodrich, J. W.; Hagstrom, T., Accurate Algorithms and Radiation Boundary Conditions for Linearized Euler Equations (1996)
[9] Goodrich, J. W.; Hagstrom, T., A Comparison of Two Accurate Boundary Treatments for Computational Aeroacoustics (1997)
[10] Grote, M. J.; Keller, J. B., Exact nonreflecting boundary conditions for the time dependent wave equation, SIAM J. Appl. Math., 55, 207 (1995) · Zbl 0817.35049
[11] Gustafsson, B.; Kreiss, H.-O.; Oliger, J., Time Dependent Problems and Difference Methods (1995)
[12] Hagstrom, T., On high-order radiation boundary conditions (1996), Springer-Verlag: Springer-Verlag New York/Berlin, p. 1
[13] Hesthaven, J. S., On the analysis and construction of perfectly matched layers for the linearized Euler equations, J. Comput. Phys., 142, 129 (1998) · Zbl 0933.76063
[14] Higdon, R. L., Initial-boundary value problems for linear hyperbolic systems, SIAM Rev., 28, 177 (1986) · Zbl 0603.35061
[15] Hixon, R.; Shih, S.-H.; Mankbadi, R. R., Evaluation of boundary conditions for computational aeroacoustics, AIAA J., 33, 2006 (1995) · Zbl 0848.76046
[16] Ho, C.-M.; Zohar, Y.; Foss, J. K.; Buell, J. C., Phase decorrelation of coherent structures in a free shear layer, J. Fluid Mech., 220, 319 (1991) · Zbl 0728.76054
[17] Hu, F. Q., Absorbing boundary conditions for linearized Euler equations by a perfectly matched layer, J. Comput. Phys., 129, 201 (1996) · Zbl 0879.76084
[18] Kreiss, H.-O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23, 277 (1970) · Zbl 0188.41102
[19] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16 (1992) · Zbl 0759.65006
[20] Lele, S. K., Computational Aeroacoustics: A Review (1997)
[21] Majda, A.; Osher, S., Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28, 607 (1975) · Zbl 0314.35061
[22] Tam, C. K.W., Advances in Numerical Boundary Conditions for Computational Aeroacoustics (1997)
[23] Tam, C. K.W.; Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107, 262 (1993) · Zbl 0790.76057
[24] Tam, C. K.W.; Webb, J. C., Radiation boundary condition and anisotropy correction for finite difference solutions of the Helmholtz equation, J. Comput. Phys., 113, 122 (1994) · Zbl 0810.65094
[25] Thompson, K. W., Time dependent boundary conditions for hyperbolic systems, J. Comput. Phys., 68, 1 (1987) · Zbl 0619.76089
[26] Trefethen, L. N., Group velocity in finite difference schemes, SIAM Rev., 24, 113 (1982) · Zbl 0487.65055
[27] Trefethen, L. N.; Halpern, L., Well-posedness of one-way wave equations and absorbing boundary conditions, Math. Comp., 47, 421 (1986) · Zbl 0618.65077
[28] Tsynkov, S. V., Numerical solution of problems on unbounded domains, Appl. Numer. Math., 27, 465 (1998) · Zbl 0939.76077
[29] Vichnevetsky, R., Wave propagation analysis of difference schemes for hyperbolic equations: A review, Int. J. Numer. Methods Fluids, 7, 409 (1987) · Zbl 0668.65063
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