×

Long-time numerical computation of wave-type solutions driven by moving sources. (English) Zbl 0987.65080

The paper is devoted to the numerical solution of the Cauchy problem \[ {\partial^2\phi\over\partial t^2}- c^2\Biggl({\partial^2\phi\over\partial x^2_1}+ {\partial^2\phi\over\partial x^2_2}+ {\partial^2\phi\over\partial x^2_3}\Biggr)= f({\mathbf x},t),\quad \phi|_{t=0}= {\partial\phi\over\partial t}\Biggl|_{t=0}= 0. \] The authors are interested in those values of \({\mathbf x}\) which belong to a ball of fixed diameter and a center moving with a certain velocity. The right-hand side \(f\) too, has the support in \({\mathbf x}\) in the same ball. Starting from any given convergent difference scheme, the authors show how to modify this procedure in order to guarantee the convergence on an arbitrarily long time interval. The paper gives numerical examples, e.g., the calculation of the acoustic field around a maneuvering aircraft.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Garabedian, P., Partial Differential Equations (1986), Chelsea: Chelsea New York · Zbl 0607.76124
[2] John, F., Partial Differential Equations (1978), Springer: Springer New York
[3] Mikhlin, S. G.; Morozov, N. F.; Paukshto, M. V., The Integral Equations of the Theory of Elasticity (1995), B.G. Teubner: B.G. Teubner Stuttgart · Zbl 0691.45003
[4] Ryaben’kii, V. S., Boundary equations with projections, Russian Math. Surv., 40, 147-183 (1985) · Zbl 0594.35035
[5] Ryaben’kii, V. S., Difference Potentials Method for Some Problems of Continuous Media Mechanics (1987), Nauka: Nauka Moscow, (in Russian)
[6] Ryaben’kii, V. S., Exact transfer of difference boundary conditions, Functional Anal. Appl., 24, 3, 251-253 (1990) · Zbl 0712.39005
[7] Ryaben’kii, V. S., Difference potentials method and its applications, Math. Nachr., 177, 251-264 (1996) · Zbl 0851.65091
[8] Ryaben’kii, V. S., Nonreflecting time-dependent boundary conditions on artificial boundaries of varying location and shape, Appl. Numer. Math., 33, 481-492 (2000) · Zbl 0967.65092
[9] Ryaben’kii, V. S.; Turchaninov, V. I.; Tsynkov, S. V., The use of lacunae of the three-dimensional wave equation for calculating the solution on long time intervals, Math. Model., 11, 12, 113-127 (1999), (in Russian) · Zbl 1189.65200
[10] V.S. Ryaben’kii, S.V. Tsynkov, V.I. Turchaninov, Global discrete artificial boundary conditions for time-dependent wave propagation, to appear as ICASE Report, NASA Langley Research Center, Hampton, VA; also submitted to J. Comput. Phys; V.S. Ryaben’kii, S.V. Tsynkov, V.I. Turchaninov, Global discrete artificial boundary conditions for time-dependent wave propagation, to appear as ICASE Report, NASA Langley Research Center, Hampton, VA; also submitted to J. Comput. Phys
[11] Tsynkov, S. V., Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27, 465-532 (1998) · Zbl 0939.76077
[12] Vladimirov, V. S., Equations of Mathematical Physics (1971), Dekker: Dekker New York · Zbl 0231.35002
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.