Geodesic
A difference scheme for the numerical solution of an advection equation with aftereffect
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2013), pp. 77-82.

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We propose a family of grid methods for the numerical solution of an advection equation with a time delay in a general form. The methods are based on the idea of separating the current state and the prehistory function. We prove the convergence of the second-order method coordinatewise and do that of the first-order with respect to time. The proof is based on techniques applied for proving analogous theorems for functional differential equations and on the general theory of difference schemes. We illustrate the obtained results with a test example.
Mots-clés : advection equations
Keywords: time delay, difference scheme, numerical methods. 
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S. I. Solodushkin. A difference scheme for the numerical solution of an advection equation with aftereffect. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2013), pp. 77-82. http://geodesic.mathdoc.fr/item/IVM_2013_10_a8/

[1] Wu J., Theory and applications of partial functional differential equations, Springer-Verlag, New York, 1996 | MR

[2] Zubik-Kowal B., Delay partial differential equations, http://www.scholarpedia.org/article/Delay_partial_differential_equations

[3] Petrov I. B., Lobanov A. I., Lektsii po vychislitelnoi matematike, Internet-universitet informatsionnykh tekhnologii, M., 2006

[4] Volkanin L. S., “Chislennoe reshenie uravneniya perenosa s effektom nasledstvennosti”, Tr. konf. “Teoriya upravleniya i matematicheskoe modelirovanie” (Izhevsk, 15–18 maya 2012), Izd. IzhGTU, Izhevsk, 2012, 12–14

[5] Karpenko D. A., Solodushkin S. I., “Chislennoe reshenie uravneniya perenosa s effektom posledeistviya”, Tez. Mezhdunarodn. 44-i Vserossiisk. molodezh. shkoly-konf. “Sovremennye problemy matematiki” (Ekaterinburg, 27 yanvarya – 2 fevralya 2013), IMM UrO RAN, Ekateriburg, 2013, 397–400

[6] Gyllenberg M., Heijmans J. A. M., “An abstract delay differential equation modelling size dependent cell growth and division”, SIAM J. Math. Anal., 18 (1987), 74–88 | DOI | MR | Zbl

[7] Rey A. O., Mackey M. C., “Multistability and boundary layer development in a transport equation with delayed arguments”, Canad. Appl. Math. Quart., 1 (1993), 61–81 | Zbl

[8] Kim A. V., Pimenov V. G., $i$-gladkii analiz i chislennye metody resheniya funktsionalno-differentsialnykh uravnenii, Regulyarnaya i khaoticheskaya dinamika, M.–Izhevsk, 2004

[9] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[10] Pimenov V. G., Lozhnikov A. B., “Raznostnaya skhema chislennogo resheniya uravneniya teploprovodnosti s posledeistviem”, Tr. IMM UrO RAN, 17, no. 1, 2011, 178–189