In this paper we investigate a mixed parabolic-hyperbolic initial boundary value problem in two disconnected intervals with Robin-Dirichlet conjugation conditions. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate is obtained.
@article{bwmeta1.element.doi-10_2478_s11533-011-0114-z,
author = {Bo\v sko Jovanovi\'c and Lubin Vulkov},
title = {Analysis and numerical approximation of a parabolic-hyperbolic transmission problem},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {73-84},
zbl = {1251.65122},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0114-z}
}
Boško Jovanović; Lubin Vulkov. Analysis and numerical approximation of a parabolic-hyperbolic transmission problem. Open Mathematics, Tome 10 (2012) pp. 73-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0114-z/
[1] Aldroubi A., Renardy M., Energy methods for a parabolic-hyperbolic interface problem arising in electromagnetism, Z. Angew. Math. Phys., 1988, 39(6), 931–936 http://dx.doi.org/10.1007/BF00945129 | Zbl 0726.35087
[2] Berres S., Bürger R., Karlsen K.H., Tory E.M., Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math., 2003, 64(1), 41–80 http://dx.doi.org/10.1137/S0036139902408163 | Zbl 1047.35071
[3] Bouziani A., Solution of a transmission problem for semilinear parabolic-hyperbolic equations by the timediscretization method, J. Appl. Math. Stoch. Anal., 2006, #61439 | Zbl 1112.35347
[4] Bramble J.H., Hilbert S.R., Bounds for a class of linear functionals with application to Hermite interpolation, Numer. Math., 1971, 16(4), 362–369 http://dx.doi.org/10.1007/BF02165007 | Zbl 0214.41405
[5] Cao Y., Yin J., Liu Q., Li M., A class of nonlinear parabolic-hyperbolic equations applied to image restoration, Nonlinear Anal. Real World Appl., 2010, 11(1), 253–261 http://dx.doi.org/10.1016/j.nonrwa.2008.11.004 | Zbl 1180.35378
[6] Datta A.K., Biological and Bioenvironmental Heat and Mass Transfer, Marcel Dekker, New York, 2002 http://dx.doi.org/10.1201/9780203910184
[7] Dupont T., Scott R., Polynomial approximation of functions in Sobolev spaces, Math. Comp., 1980, 34(150), 441–463 http://dx.doi.org/10.1090/S0025-5718-1980-0559195-7 | Zbl 0423.65009
[8] Gegovska-Zajkova S., Jovanovic B.S., Jovanovic I.M., On the numerical solution of a transmission eigenvalue problem, Lecture Notes in Comput. Sci., 5434, Springer, Berlin, 2009, 289–297 | Zbl 1233.65083
[9] Givoli D., Exact representation on artificial interfaces and applications in mechanics, Applied Mechanics Reviews, 1999, 52(11), 333–349 http://dx.doi.org/10.1115/1.3098920
[10] Jovanović B.S., The Finite Difference Method for Boundary-Value Problems with Weak Solutions, Posebna Izdan., 16, Matematički Institut u Beogradu, Belgrade, 1993
[11] Jovanović B.S., Vulkov L.G., Numerical solution of a hyperbolic transmission problem, Comput. Methods Appl. Math., 2008, 8(4), 374–385 | Zbl 1202.65111
[12] Jovanović B.S., Vulkov L.G., Numerical solution of a two-dimensional parabolic transmission problem, Int. J. Numer. Anal. Model., 2010, 7(1), 156–172
[13] Jovanović B.S., Vulkov L.G., Numerical solution of a parabolic transmission problem, IMA J. Numer. Anal., 2011, 31(1), 233–253 http://dx.doi.org/10.1093/imanum/drn077 | Zbl 1215.65141
[14] Korzyuk V.I., A conjugacy problem for equations of hyperbolic and parabolic types, Differencial’nye Uravnenija, 1968, 4(10), 1855–1866 (in Russian) | Zbl 0165.11201
[15] Korzyuk V.I., Lemeshevsky S.V., Problems on conjugation of polytypic equations, Math. Model. Anal., 2001, 6(1), 106–116 | Zbl 1003.35100
[16] Lions J.-L., Magenes E., Non-Homogeneous Boundary Value Problems and Applications. I, Grundlehren Math. Wiss., 181, Springer, Berlin-Heidelberg-New York, 1972
[17] Mascia C., Porretta A., Terracina A., Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations, Arch. Ration. Mech. Anal., 2002, 163(2), 87–124 http://dx.doi.org/10.1007/s002050200184 | Zbl 1027.35081
[18] Oganesyan L.A., Rukhovets L.A., Variational-Difference Methods for Solving Elliptic Equations, Akad. Nauk Armyan. SSR, Erevan, 1979 (in Russian) | Zbl 0496.65053
[19] Qin Y., Nonlinear Parabolic-Hyperbolic Coupled Systems and their Attractors, Oper. Theory Adv. Appl., 184, Birkhäuser, Basel, 2008
[20] Rogov B.V., Hyperbolic-parabolic approximation of the Reynolds equations for turbulent flows of chemically reacting gas mixtures, Mat. Model., 2004, 16(12), 20–39 (in Russian) | Zbl 1097.76585
[21] Samarskii A.A., The Theory of Difference Schemes, Monogr. Textbooks Pure Appl. Math., 240, Marcel Dekker, New York, 2001 http://dx.doi.org/10.1201/9780203908518
[22] Samarskii A.A., Korzyuk V.I., Lemeshevsky S.V., Matus P.P., Difference schemes for the conjugation problem of hyperbolic and parabolic equations on moving grids, Dokl. Akad. Nauk, 1998, 361(3), 321–324 (in Russian)
[23] Samarskii A.A., Korzyuk V.I., Lemeshevsky S.V., Matus P.P., Finite-difference methods for problem of conjugation of hyperbolic and parabolic equations, Math. Models Methods Appl. Sci., 2000, 10(3), 361–377 http://dx.doi.org/10.1142/S0218202500000227 | Zbl 1012.65087
[24] Samarskii A.A., Lazarov R.D., Makarov V.L., Difference Schemes for Differential Equations with Generalized Solutions, Vysshaya Shkola, Moscow, 1987 (in Russian)
[25] Wloka J., Partial Differential Equations, Cambridge University Press, Cambridge, 1987