Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)13387Research ArticleSolution of the Boundary-Value Problem for a Systems of ODEs of Large Dimension: Benchmark Calculations in the Framework of Kantorovich MethodGusevA Agooseff@jinr.ruChuluunbaatarOchuka@jinr.ruVinitskyS IRUDN University, Moscow, Russiavinitsky@theor.jinr.ruDerbovV Lderbov@sgu.ruJoint Institute for Nuclear ResearchSaratov State University150320163313717092016Copyright © 2016,2016We present benchmark calculations of the boundary-value problem (BVP) for a systems of second order ODEs of large dimension with help of KANTBP program using a finite element method. In practice, for solving the BVPs with the long-range potentials and a large number of open channels there is a necessity of solving boundary value problems of the large-scale systems of differential equations that require further investigation of convergence and stability of the algorithms and programs. With this aim we solve here the eigenvalue problem for an elliptic differential equation in a two-dimensional domain with Dirichlet boundary conditions. The solution is sought in the form of Kantorovich expansion over the parametric basis functions of one of the independent variables with the second variable treated as a parameter. The basis functions are calculated in an analytical form as solutions of the auxiliary parametric Sturm-Lioville problem for a second-order ODE. As a result, the two-dimensional problem is reduced to a boundary-value problem for a set of self-adjoint second-order ODEs for functions of the second independent variable. The discrete formulation of the problem is implemented using the finite element method. The efficiency, stability and convergence of the calculation scheme is shown by benchmark calculations for a triangle membrane with a degenerate spectrum.benchmark calculationsboundary-value problemlarge-scale systems of ODEsKantorovich methodfinite element methodтестовые расчетыкраевая задачасистемы ОДУ большой размерностиметод Канторовичаметод конечных элементов[Metastable States of a Composite System Tunneling Through Repulsive Barriers / A.A. Gusev, S.I. Vinitsky, O. Chuluunbaatar, V.L. Derbov, A. G´o´zd´z, P. M. Krassovitskiy // Theoretical and Mathematical Physics. - 2016. - Vol. 186. - Pp. 21-40.][Symbolic-Numeric Algorithms for Computer Analysis of Spheroidal Quantum Dot Models / A.A. Gusev, O. Chuluunbaatar, V.P. Gerdt, V.A. Rostovtsev, S.I. Vinitsky, V.L. Derbov, V.V. Serov // Lecture Notes in Computer Science. - 2010. - Vol. 6244. - Pp. 106-122.][On Calculations of Two-Electron Atoms in Spheroidal Coordinates Mapping on Hypersphere / S.I. Vinitsky, A.A. Gusev, O. Chuluunbaatar, V.L. Derbov, A.S. Zotkina // Proc. SPIE. - 2016. - Vol. 9917. - P. 99172Z.][ODPEVP: A Program for Computing Eigenvalues and Eigenfunctions and their First Derivatives with Respect to the Parameter of the Parametric Self-Adjoined Sturm- Liouville Problem / O. Chuluunbaatar, A. A. Gusev, S. I. Vinitsky, A. G. Abrashkevich // Comput. Phys. Commun. - 2009. - Vol. 180. - Pp. 1358-1375.][POTHEA: A Program for Computing Eigenvalues and Eigenfunctions and Their First Derivatives with Respect to the Parameter of the Parametric Self-Adjoined 2D Elliptic Partial Diﬀerential Equation / A. A. Gusev, O. Chuluunbaatar, S. I. Vinitsky, A.G. Abrashkevich // Comput. Phys. Commun. - 2014. - Vol. 185. - Pp. 2636- 2654.][KANTBP 2.0: New Version of a Program for Computing Energy Levels, Reaction Matrix and Radial Wave Functions in the Coupled-Channel Hyperspherical Adiabatic Approach / O. Chuluunbaatar, A. A. Gusev, S. I. Vinitsky, A. G. Abrashkevich // Comput. Phys. Commun. - 2008. - Vol. 179. - Pp. 685-693.][Kantorovich L.V., Krylov V.I. Approximate Methods of Higher Analysis. - New York: Wiley, 1964.][Strang G., Fix G. J. An Analysis of the Finite Element Method. - New York: Prentice-Hall, Englewood Cliﬀs, 1973.][Pockels F. ¨ Uber die partielle Diﬀerential-Gleichung Δ+2 = 0 und deren auftreten in der mathematischen physik. - Leipzig: B. G. Teubner, 1891.][Solution of Boundary-Value Problems using Kantorovich Method / A.A. Gusev, L.L. Hai, O. Chuluunbaatar, S.I. Vinitsky, V.L. Derbov // EPJ Web of Conferences. - 2016. - Vol. 108. - P. 02026.]