Abstract

We consider a firstorder 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 finitedimensional and infinitedimensional state components. These grid methods are: analog of running schemes family, analog of CrankNicolson scheme, an approximation method to the middle of the square. The onedimensional 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 functionaldifferential equations, theorems of orders of proposed algorithms convergence are received. Test examples comparing errors of methods are given.

References

 Wu J. Theory and application of partial functional differential equations, New York: SpringerVerlag, 1996, 438 p.
 Samarskii А.А. Teoriya raznostnykh skhem (Theory of difference schemes), Moscow: Nauka, 1989, 656 p.
 Kalitkin N.N. Chislennye metody (Numerical methods), St. Petersburg: BHVPetersburg, 2011, 586 p.
 Petrov I.B., Lobanov A.I. Lektsii po vychislitel'noi matematike (Lectures on calculus mathematics), Moscow: Binom, 2006, 524 p.
 Kamont Z., Czernous W. Implicit difference methods for HamiltonJacobi functional differential equations, Numerical Analysis and Applications, 2009, vol. 2, no. 1, pp. 4657.
 Volkanin L.S. Numerical solution of advection equations with delay, Teoriya upravleniya i matematicheskoe modelirovanie: tez. dokl. Vserossiiskoi konferentsii (Theory of control and mathematical modeling: abstracts of AllRussian conference), Izhevsk Technical State University, Izhevsk, 2012, pp. 1213 (in Russian).
 Volkanin L.S., Pimenov V.G. Grid schemes for the solution of the equation of advection with delay, Aktual'nye problemy prikladnoi matematiki i mekhaniki: tez. dokl. VI Vserossiiskoi konferentsii (Actual problems of applied mathematics and mechanics: abstracts of VI AllRussian conference dedicated to memory of the academician A.F.Sidorov), Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, AbrauDurso, 2012, pp. 2223 (in Russian).
 Solodushkin S.I. A difference scheme for the numerical solution of an advection equation with aftereffect, Russian Mathematics, 2013, vol. 57, no. 10, pp. 6570.
 Kim A.V., Pimenov V.G. igladkii analiz i chislennye metody resheniya funktsional'nodifferentsial'nykh uravnenii (ismooth calculus and numerical methods for functional differential equations), MoscowIzhevsk: Regular and Chaotic Dynamics, 2004, 256 p.
 Pimenov V.G. General linear methods for the numerical solution of functionaldifferential equations, Differential Equations, 2001, vol. 37, no. 1, pp. 116127.
 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, 2011, vol. 17, no. 1, pp. 178189 (in Russian).
