Geodesic
Grid methods of solving advection equations with delay
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2014), pp. 59-74 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a first-order partial differential equation with heredity effect $$ \frac{\partial u(x,t)}{\partial t}+a\frac{\partial u(x,t)}{\partial x}=f(x,t,u(x,t),u_t(x,\cdot)),\quad u_t(x,\cdot)=\{u(x,t+s),\ -\tau\leqslant s<0\}. $$ For such an equation we construct grid methods using the principle of separation of finite-dimensional and infinite-dimensional state components. These grid methods are: analog of running schemes family, analog of Crank–Nicolson scheme, an approximation method to the middle of the square. The one-dimensional and double piecewise linear interpolation and the extrapolation by continuation are applied in order to account the effect of heredity. It is shown that the considered methods have orders of a local error: $O(h+\Delta)$, $O(h+\Delta^2)$ and $O(h^2+\Delta^2)$ respectively, where $h$ is the spatial discretization interval, $\Delta$ is the time discretization interval. Properties of double piecewise linear interpolation are investigated. Using the results of the general theory of differential schemes, stability conditions of the proposed methods are established. Including them in the general scheme of numerical methods for the functional-differential equations, theorems of orders of proposed algorithms convergence are received. Test examples comparing errors of methods are given.
Mots-clés : advection equation, interpolation, extrapolation, convergence order.
Keywords: delay, grid schemes, stability 
@article{VUU_2014_3_a5,
     author = {V. G. Pimenov and S. V. Sviridov},
     title = {Grid methods of solving advection equations with delay},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {59--74},
     year = {2014},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2014_3_a5/}
}
TY  - JOUR
AU  - V. G. Pimenov
AU  - S. V. Sviridov
TI  - Grid methods of solving advection equations with delay
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2014
SP  - 59
EP  - 74
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VUU_2014_3_a5/
LA  - ru
ID  - VUU_2014_3_a5
ER  - 
%0 Journal Article
%A V. G. Pimenov
%A S. V. Sviridov
%T Grid methods of solving advection equations with delay
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2014
%P 59-74
%N 3
%U http://geodesic.mathdoc.fr/item/VUU_2014_3_a5/
%G ru
%F VUU_2014_3_a5
V. G. Pimenov; S. V. Sviridov. Grid methods of solving advection equations with delay. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2014), pp. 59-74. http://geodesic.mathdoc.fr/item/VUU_2014_3_a5/

[1] Wu J., Theory and application of partial functional differential equations, Springer-Verlag, New York, 1996, 438 pp. | MR

[2] Samarskii A. A., Theory of difference schemes, Nauka, Moscow, 1989, 656 pp. | MR

[3] Kalitkin N. N., Numerical methods, BHV-Petersburg, St. Petersburg, 2011, 586 pp.

[4] Petrov I. B., Lobanov A. I., Lectures on calculus mathematics, Binom, Moscow, 2006, 524 pp.

[5] Kamont Z., Czernous W., “Implicit difference methods for Hamilton–Jacobi functional differential equations”, Numerical Analysis and Applications, 2:1 (2009), 46–57 | DOI | Zbl

[6] Volkanin L. S., “Numerical solution of advection equations with delay”, Theory of control and mathematical modeling, Abstracts of All-Russian conference, Izhevsk Technical State University, Izhevsk, 2012, 12–13 (in Russian)

[7] Volkanin L. S., Pimenov V. G., “Grid schemes for the solution of the equation of advection with delay”, Actual problems of applied mathematics and mechanics, Abstracts of VI All-Russian conference dedicated to memory of the academician A. F. Sidorov, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Abrau-Durso, 2012, 22–23 (in Russian)

[8] Solodushkin S. I., “A difference scheme for the numerical solution of an advection equation with aftereffect”, Russian Mathematics, 57:10 (2013), 65–70 | DOI | Zbl

[9] Kim A. V., Pimenov V. G., $i$-smooth calculus and numerical methods for functional differential equations, Regular and Chaotic Dynamics, Moscow–Izhevsk, 2004, 256 pp.

[10] Pimenov V. G., “General linear methods for the numerical solution of functional-differential equations”, Differential Equations, 37:1 (2001), 116–127 | DOI | MR | Zbl

[11] Pimenov V. G., Lozhnikov A. B., “Difference schemes for the numerical solution of the heat conduction equation with aftereffect”, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 17, no. 1, 2011, 178–189 (in Russian) | Zbl