Abstract
Initial-boundary value problems formulated in spatially unbounded domains can be sometimes reduced to problems in their bounded subdomains by using the so-called open boundary conditions. These conditions are set on the surface separating the subdomain from the rest of the domain. One of the approaches to obtaining such a kind of conditions is based on an approximation of the kernels of the time convolution operators in the relations connecting the exact solution of the original problem and its derivatives on the open boundary. In this case, it is possible to considerably reduce the requirements for system resources required to solve numerically for a wide range of physical and engineering problems. Estimates of the perturbations of the exact solution due to the approximate conditions are obtained for a model problem with one space variable.
Similar content being viewed by others
References
Yu. V. Bobylev, M. V. Kuzelev, A. A. Rukhadze, and A. G. Sveshnikov, “Nonstationary Partial Radiation Conditions in Relativistic High-Current Plazma VHF Electronics,” Fiz. Plazmy 25(7), 615–620 (1999).
N. S. Ginzburg, N. A. Zavol’skii, G. S. Nusinovich, and A. S. Sergeev, “Establishment of Self-Oscillations in Electronic VHF Generators with a Diffraction Coupling-Out of Radiation,” Izv. Vyssh. Uchebn. Zaved., Radiofiz. 29(1), 106–114 (1986).
M. V. Urev, “Boundary Conditions for Maxwell Equations with Arbitrary Time Dependence,” Zh. Vychisl. Mat. Mat. Fiz. 37, 1489–1497 (1997) [Comput. Math. Math. Phys. 37, 1444–1451 (1997)].
A. R. Maikov and A. G. Sveshnikov, “Radiation Conditions for Discrete Analogs of Time-Dependent Maxwell’s Equations in the Case of a Inhomogeneous Medium,” Zh. Vychisl. Mat. Mat. Fiz. 35, 412–426 (1995).
A. Arnold and M. Ehrhardt, “Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equation in Underwater Acoustics,” J. Comput. Phys. 145, 611–638 (1998).
T. Hagstrom, “Radiation Boundary Conditions for the Numerical Simulation of Waves,” Acta Numerica 6, 47–106 (1999).
T. Hagstrom, “New Results on Absorbing Layers and Radiation Boundary Conditions,” Lect. Notes Comput. Sci. Eng. 31, 1–42 (2003).
S. V. Tsynkov, “Numerical Solution of Problems on Unbounded Domains. A Review,” Appl. Numer. Math. 27, 465–532 (1998).
D. Givoli and B. Neta, High-order Higdon Non-Reflecting Boundary Conditions for the Shallow Water Equations,” NPS-MA-02-001 (Naval Postgraduate School, Monterey, CA, 2002).
D. Givoli and B. Neta, “High-Order Non-Reflecting Boundary Conditions for Dispersive Waves,” Wave Motion 37, 257–271 (2003).
Yu. K. Sirenko, V. P. Shestopalov, and N. P. Yashina, “New Methods in the Dynamic Linear Theory of Open Waveguide Resonators,” Zh. Vychisl. Mat. Mat. Fiz. 37, 869–877 (1997) [Comput. Math. Math. Phys. 37, 845–853 (1997)].
B. Alpert, L. Greengard, and T. Hagstrom, “Nonreflecting Boundary Conditions for the Time-Dependent Wave Equations,” J. Comput. Phys. 180, 270–296 (2002).
I. L. Sofronov, “Non-Reflecting Inflow and Outflow in Wind Tunnel for Transonic Time-Accurate Simulation,” J. Math. Anal. Appl. 221, 92–115 (1998).
A. R. Maikov and A. G. Sveshnikov, “On Rigorous and Approximate Nonstationary Partial Radiation Conditions,” J. Commun. Technol. Electronics 45(Suppl. 2), 196–211 (2000).
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer, New York, 1985).
O. A. Ladyzhenskaya, Mixed Problem for a Hyperbolic Equation (Gostekhteorizdat, Moscow, 1953) [in Russian].
G. N. Watson, A Treatise on the Theory of Bessel Functions, Vol. 1 (Cambridge Univ. Press, Cambridge, 1945; Inostrannaya Literature, Moscow 1949).
L. V. Kantorovich and G. P. Akilov, Functional Analysis (3rd ed., Nauka, Moscow, 1984; Pergamon Press, Oxford, 1982).
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations (Inostrannaya Literature, Moscow, 1960; Clarendon, Oxford, 1962).
A. R. Maikov, A. G. Sveshnikov, and S. A. Yakunin, “A Finite Difference Scheme for Time-Dependent Maxwell’s Equations in Waveguide Systems,” Zh. Vychisl. Mat. Mat. Fiz. 26, 851–863 (1986).
V. S. Vladimirov, Distributions in Mathematical Physics (Nauka, Moscow, 1976) [in Russian].
N. I. Akhiezer, Lectures on the Theory of Approximation, 2nd ed., (Ungar, New York, 1956; Nauka, Moscow, 1965).
A. R. Maikov, “On Approximate Open Boundary Conditions for the Wave Equation and the Klein-Gordon Equation,” Vychisl. Metody Programmirovanie 6, 290–303 (2005); http://num-meth.srcc.msu.su.
Author information
Authors and Affiliations
Additional information
Original Russian Text © A.R. Maikov, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 6, pp. 1058–1073.
Rights and permissions
About this article
Cite this article
Maikov, A.R. Approximate open boundary conditions for a class of hyperbolic equations. Comput. Math. and Math. Phys. 46, 1007–1022 (2006). https://doi.org/10.1134/S0965542506060091
Received:
Issue Date:
DOI: https://doi.org/10.1134/S0965542506060091