V.G. Pimenov, A.B. Lozhnikov. Richardson method for a diffusion equation with functional delay ... P. 133-144

A diffusion equation with a functional delay effect is considered. The problem is discretized. Constructions of the Crank–Nicolson difference method with piecewise linear interpolation and extrapolation by continuation are given; the method here has the second order of smallness with respect to the sampling steps in time $\Delta$ and space $h$. The basic Crank–Nicolson method with piecewise cubic interpolation and extrapolation by continuation is constructed. The order of the residual without interpolation of the base method is studied, and the expansion coefficients of the residual with respect to $\Delta$ and $h$ are written. An equation for the leading term of the asymptotic expansion of the global error is written. Under certain assumptions, the validity of the application of the Richardson extrapolation procedure is substantiated and an appropriate method is constructed. The main of these assumptions is the consistency of the orders of smallness of $\Delta$ and $h$. It is proved that the method has order $O(\Delta^4+h^4)$. The results of numerical experiments on test examples are presented.

Keywords: diffusion equation, functional delay, Crank–Nicolson method, piecewise cubic interpolation, extrapolation by continuation, Richardson method

Received March 14, 2023

Revised April 10, 2023

Accepted April 17, 2023

Funding Agency: This work was supported by the Russian Science Foundation (project no. 22-21-00075).

Vladimir Germanovich Pimenov, Dr. Phys.-Math. Sci., Prof., Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: v.g.pimenov@urfu.ru

Andrey Borisovich Lozhnikov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: ablozhnikov@yandex.ru


1.   Wu J. Theory and application of partial functional differential equations, NY, Springer-Verlag, 1996, 432 p. doi: 10.1007/978-1-4612-4050-1

2.   Kamont Z., Kropielnicka K. Implicit difference methods for evolution functional differential equations. Numer. Anal. Appl., 2011, vol. 4, no. 4, pp. 294–308. doi: 10.1134/S1995423911040033

3.   Pimenov V.G., Lozhnikov A.B. Difference schemes for the numerical solution of the heat conduction equation with aftereffect. Proc. Steklov Inst. Math., 2011, vol. 275, no. 1, pp. S137–S148. doi: 10.1134/S0081543811090100

4.   Pimenov V.G. Raznostnye metody resheniya uravnenii v chastnych proizvodnykh s nasledstvennost’u [Difference methods of solving partial differential equations with heredity]. Yekaterinburg, Ural State Univ. publ., 2014, 134 p. ISBN: 978-5-7996-1364-8.

5.   Sun Z., Zhang Z. A linearized compact difference scheme for a class of nonlinear delay partial differetial equations. Appl. Math. Model., 2013, vol. 37, no. 3, pp. 742–752. doi: 10.1016/j.apm.2012.02.036

6.   Li D., Zhang C., Wen J. A note on compact finite difference method for reaction-diffusion equations with delay. Appl. Math. Model., 2015, vol. 39, no. 5–6, pp. 1749–1754. doi: 10.1016/j.apm.2014.09.028

7.   Amiraliyev G.M., Cimen E., Amirali I., Cakir M. High-order finite difference technique for delay pseudo-parabolic equations. J. Comput. Appl. Math., 2017, vol. 321, pp. 1–7. doi: 10.1016/j.cam.2017.02.017

8.   Wang W., Rao W., Zhong P. A posteriori error analysis for Crank-Nicolson-Galerkin type methods for reaction-diffusion equations with delay. SIAM J. Sci. Comput., 2018, vol. 40, no. 2, pp. A1095–A1120. doi: 10.1137/17M1143514

9.   Marchuk G.I., Shaidurov V.V. Povyshenie tochnosti reshenii raznostnykh skhem [Improving the accuracy of finite difference schemes] Moscow, Nauka Publ., 1979, 320 p.

10.   Hairer E., Nјrsett S.P., Wanner G. Solving ordinary differential equations I. Nonstiff problems, Berlin, Heidelberg, Springer, 1987, 482 p. doi: 10.1007/978-3-662-12607-3 Translated to Russian under the title Reshenie obyknovennykh differentsial’nykh uravnenii. Nezhestkie zadachi, Moscow, Mir Publ., 1990, 510 p. ISBN: 5-03-001179-X.

11.   Qian L.Z., Gu H.B. High order compact scheme combined with extrapolation technique for solving convection-diffusion equations. J. Shandong Univ. Nat. Sci., 2011, vol. 46, no. 12, pp. 39-43.

12.   Zhang Q., Zhang C. A compact difference scheme combined with extrapolation techniques for solving a class of neutral delay parabolic differential equations. Appl. Math. Letters, 2013, vol. 26, no. 2, pp. 306–312. doi: 10.1016/j.aml.2012.09.015

13.   Zhang C., Tan Z. Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations. Commun. Nonlinear Sci. and Numer. Simul., 2020, vol. 91, article no. 105461. doi: 10.1016/j.cnsns.2020.105461

14.   Deng D., Chen J. Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays. Networks and Heterogeneous Media, 2023, vol. 18, no. 1, pp. 412–443. doi: 10.3934/nhm.2023017

15.   Kim A.V., Pimenov V.G. i-gladkii analiz i chislennye metody resheniya funktsional’no-differentsial’nykh uravnenii [i-smooth analysis and numerical methods of solving functional-differential equations], Moscow, Izhevsk, Regular and Chaotic Dynamics, 2004, 256 p.

16.   Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal control, N.Y., Springer, 1987, 309 p. doi: 10.1007/978-1-4615-7551-1. Original Russian text was published in Alekseev V.M., Tikhomirov V.M., Fomin S.V., Optimal’noe upravlenie, Moscow, Nauka Publ., 1979, 426 p.

17.   Programs for Solving Partial Differential Equations with Delay [e-resource]. 2023. Available on: https://github.com/PDDEsoft/Parabolic 

Cite this article as: V.G. Pimenov, A.B. Lozhnikov. Richardson method for a diffusion equation with functional delay. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 133–144; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S204–S215.