519.62 Application of the one–step Galerkin method for solving a system of ordinary differential equations with initial conditions

Russikikh S. V. (Moscow Aviation Institute (National Research University)), Shklyarchuk F. N. (Moscow Aviation Institute (National Research University))

ORDINARY DIFFERENTIAL EQUATIONS, NONLINEAR SYSTEMS, INITIAL PROBLEM, NUMERICAL SOLUTIONS, ONE-STEP GALERKIN METHOD

doi: 10.18698/2309-3684-2022-3-1832

A nonlinear oscillatory system described by ordinary differential equations with variable coefficients is considered. It is assumed that in the time interval under consideration, the solution of the system is sufficiently smooth - without discontinuities, collisions and bifurcations. From an inhomogeneous system of equations, terms that depend linearly on coordinates, velocities and accelerations and terms that depend non-linearly on these variables are explicitly distinguished. A new approach is proposed for numerical solution by the step method of the initial problem described by such a system of ordinary differential equations of the second order. At the integration step, unknown functions are represented as a sum of functions satisfying the initial conditions: a linear Euler solution and several given correcting functions in the form of polynomials of the second and higher degrees with unknown coefficients. The differential equations at the step are satisfied approximately in the sense of a weak solution by the Galerkin method on a system of corrective functions. Algebraic equations with nonlinear terms are obtained, which are solved by iteration, starting in the first approximation with a linear solution. The resulting solution at the end of this step is used as the initial conditions for the next step. As an example, we consider one homogeneous second-order differential equation without the first derivative with strong cubic nonlinearity in coordinate (at maximum amplitude, the nonlinear force is twice the linear force). This equation has an exact periodic solution in the form of an integral of the energy of a conservative system, which is used to estimate the accuracy of numerical solutions obtained by Galerkin, Runge-Kutta and Adams methods of the second order, as well as by Radau5 and BDF methods at various time intervals (up to 8000 periods of free oscillations of the system) using various constant integration steps (from 0.0025 fractions of a period). At the same time, in the Galerkin method, four identical correction functions were used at each step in the form of polynomials from the second to the fifth degree. It is shown that for large time intervals of calculations, the Galerkin method has a higher accuracy compared to other numerical methods considered. Therefore, it can be used for the numerical solution of nonlinear problems in which it is required to solve them over long time intervals; for example, when calculating steady-state limit cycles of nonlinear oscillations and chaotic nonlinear oscillations with strange attractors.

Demidovich B.P., Maron I.A., Shuvalova E.Z. Chislennye metody analiza [Numerical methods of analysis]. Moscow, Nauka Publ., 1967, 368 p.
Bakhvalov I.V., Zhidkov N.P., Kobelkov G.M. Chislennye metody [Numerical methods]. Moscow, Laboratory of Basic Knowledge Publ., 2000, 630 p.
Collatz L. Chislennye metody resheniya differencial'nyh uravnenij [Numerical methods for solving differential equations]. Moscow, Foreign Literature Publ., 1953, 460 p.
Hairer E., Nersett S., Wanner G. Reshenie obyknovennyh differencial'nyh uravnenij. Nezhestkie zadachi [Solve ordinary differential equations. Nonrigid problems]. Moscow, Mir Publ., 1990, 512 p.
Hairer E., Wanner G. Reshenie obyknovennyh differencial'nyh uravnenij. Zhestkie i differencial'no–algebraicheskie zadachi [Solve ordinary differential equations. Rigid and differential–algebraic problems]. Moscow, Mir Publ., 1999, 685 p.
Skvortsov L.M. Explicit multistep method for the numerical solution of stiff differential equations. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 6, pp. 915–923.
Latypov A.F., Nikulichev Yu.V. Numerical methods based on multipoint Hermite interpolating polynomials for solving the cauchy problem for stiff systems of ordinary differential equations. Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 2, pp. 227–237.
Bulatov M.V., Tygliyan A.V., Filippov S.S. A class of one-step one-stage methods for stiff systems of ordinary differential equations. Computational Mathematics and Mathematical Physics, 2011, vol. 51, no. 7, pp. 1167–1180.
Skvortsov L.M. A fifth order implicit method for the numerical solution of differential–algebraic equations. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 6, pp. 962–968.
Belov A.A., Kalitkin N.N. Curvature-based grid step selection for stiff Cauchy problems. Mathematical Models and Computer Simulations, 2017, vol. 9, no. 3, pp. 305–317.
Volkov-Bogorodskii D.B., Danilin A.N., Kuznetsov E.B., Shalashilin V.I. Implicit methods for integration of initial value problems for parameterized systems of second-order ordinary differential equations. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 11, pp. 1620–1631.
Bulatov M.V., Gorbunov V.K., Martynenko Yu.V., Kong N.D. Variational approaches to numerical solution of differential algebraic equations. Computational Technologies, 2010, vol. 15, no. 5, pp. 3–13.
Chistyakov V.F., Chistyakova E.V. Application of the least squares method to solving linear differential-algebraic equations. Numerical Analysis and Applications, 2013, vol. 6, no. 1, pp. 77-90.
Zaletkin S.F. Numerical integration of ordinary differential equations using orthogonal expansions. Matematicheskoe modelirovanie [Mathematical modeling], 2010, vol. 22, no. 1, pp. 69–85.
Arushanyan O.B., Zaletkin S.F. On some analytic method for approximate solution of systems of second order ordinary differential equations. MoscowUniversity Mathematics Bulletin, 2019, vol. 74, no. 3, pp. 127–130.
Aulchenko S.M., Latypov A.F., Nikulichev Yu.V. Metod chislennogo integrirovaniya sistem obyknovennyh differencial'nyh uravnenij s ispol'zovaniem interpolyacionnyh polinomov Ermita [Method of numerical integration of systems of ordinary differential equations using Hermite interpolation polynomials]. Zhurnal vychislitel'noj matematiki i matematicheskoj fiziki [Journal of Computational Mathematics and Mathematical Physics], 1998, vol. 38, no.10, pp.1665–1670.
Latypov A.F., Popik O.V. Computational method for solving of the cauchyproblem for stiff systems of ordinary differential equations based on multilinkinterpolated hermite polynomials. Computational Technologies, 2011, vol. 16, no. 2, pp. 78–85.
Ershov N.F., Shakhverdi G.G. Metod konechnyh elementov v zadachah gidrodinamiki i gidrouprugosti [Finite element method in problems of hydrodynamics and hydroelasticity]. Leningrad, Shipbuilding Publ.. 1984, 240 p.
Ioriatti M., Dumbser M. Semi-implicit staggered discontinuous Galerkinschemes for axially symmetric viscous compressible flows in elastic tube. Computers and Fluids, 2018, vol.167, pp. 166–179.
Kulikov G.Y., Khrustaleva E.Y. Automatic step size and order control in onestep collocation methods with higher derivatives. Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 6, pp. 1006–1023.
Weiner R., Kulikov G.Y. Efficient error control in numerical integration of ordinary differential equations and optimal interpolating variable-stepsize peermethods. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no 4, pp. 591–607.
Bazhenov V.G., Chekmarev D.T. Reshenie zadach nestacionarnoj dinamiki plastin i obolochek variacionno-raznostnym metodom [Solving problems of unsteady dynamics of plates and shells by the variational-difference method]. Nizhny Novgorod, UNN Publ., 2000, 118 p.
Belytschko T. A survey of numerical methods and computer programs for dynamic structural analysis. Nuclear Engineering and Design, 1976, vol. 37, I ss. 1, pp. 23–34.
Houbolt J.C. A recurrence matrix solution for the dynamic response of elastic aircraft. Journal of Aeronautical Sciences, 1950, vol. 17, pp. 540–550.
Newmark N.M. A method of computation for structural dynamics. ASCE Journal of the Engineering Mechanics Division, 1959, vol. 85, pp. 67–94.
Bathe K.J., Wilson E.L. Numerical methods in finite element analysis. New York, Prentice-Hall, 1976, 544 p.
Krieg R.D., Key S.W. Transient shell response by numerical time integration. International Journal for Numerical Methods in Engineering, 1973, vol. 7, no. 3, pp. 273–286

Русских С.В., Шклярчук Ф.Н. Применение одношагового метода Галеркина для решения системы обыкновенных дифференциальных уравнений с начальными условиями. Математическое моделирование и численные методы, 2022, № 3, с. 18–32.

Работа выполнена при финансовой поддержке РНФ (проект № 22-29-01206).