The convergence of approximate solutions of the Cauchy problem for the model of quasi-steady process in conducting nondispersive medium with relaxation. (English) Zbl 1359.65141

Bogatyreva, E. A.

J. Comput. Eng. Math. 3, No. 2, 25-31 (2016).

Author’s abstract: This article deals with numerical method for solving of the Dirichlet-Cauchy problem for equation modeling the quasi-steady process in conducting nondispersive medium with relaxation. This problem describes a complex electrodynamic process, allows to consider and predict its flow in time. The study of electrodynamic models is necessary for the development of electrical engineering and new energy saving technologies. The main equation of the model is considered as a quasi-linear Sobolev type equation. The convergence of approximate solutions obtained from the use of the method of straight lines with \(\varepsilon\)-embedding method and complex Rosenbrock method is proven in the article. The lemmas on the local error and on the distribution of error are proven. Estimates of a global error of the method are obtained.

Reviewer: Maria Gousidou-Koutita (Thessaloniki)

MSC:

65L80	Numerical methods for differential-algebraic equations
65L05	Numerical methods for initial value problems involving ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations
65J15	Numerical solutions to equations with nonlinear operators

Keywords:

Rosenbrock method; quasi-linear Sobolev type equation; weak generalized solution; numerical solution

References:

[1]	[1] М.О. Корпусов, Ю.Д. Плетнер, А.Г. Свешников, ”О квазистационарных процессах в проводящих средах без дисперсии”, Ж. вычисл. матем. и матем. физ., 40:8 (2000), 1237–1249 · Zbl 0993.78013
[2]	[2] М.О. Корпусов, ””Разрушение” решения псевдопараболического уравнения с производной по времени от нелинейного эллиптического оператора”, Ж. вычисл. матем. и матем. физ., 42:12 (2002), 1788–1795 · Zbl 1116.35336
[3]	[3] А.Г. Свешников, A.Б. Альшин, М.О. Корпусов, Нелинейный функциональный анализ и его приложения к уравнениям в частных производных, Науч. мир, Москва, 2008, ISBN: 978-5-91522-011-8, 399 с. · Zbl 0342.02023
[4]	[4] E. A. Bogatyreva, ”Numerical modeling of quasi-steady process in conducting nondispersive medium with relaxation”, J. Comp. Eng. Math., 2:1 (2015), 45–51 · Zbl 1338.78021 · doi:10.14529/jcem150105
[5]	[5] А.Б. Альшин, Е.А. Альшина, Н.Н. Калиткин, А.Б. Корягина, ”Схемы Розенброка с комплексными коэффициентами для жестких и дифференциально-алгебраических систем”, Ж. вычисл. матем. и матем. физ., 46:8 (2006), 1392–1414 · Zbl 1204.11080 · doi:10.1134/S0965542506080057