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Title: Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods

Abstract

We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Nørsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numericalmore » and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.« less

Authors:
 [1]
  1. Khristianovich Institute of Theoretical and Applied Mechanics of Siberian Branch of RAS (Russian Federation)
Publication Date:
OSTI Identifier:
22773577
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Sciences
Additional Journal Information:
Journal Volume: 236; Journal Issue: 2; Other Information: Copyright (c) 2019 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; 42 ENGINEERING; ACCURACY; ANALYTICAL SOLUTION; BOUNDARY-VALUE PROBLEMS; COMPARATIVE EVALUATIONS; EQUATIONS; LEGENDRE POLYNOMIALS; ONE-DIMENSIONAL CALCULATIONS; RUNGE-KUTTA METHOD; STABILITY; SYMMETRY; THERMAL CONDUCTION

Citation Formats

Yankovskii, A. P. Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods. United States: N. p., 2019. Web. doi:10.1007/S10958-018-4101-9.
Yankovskii, A. P. Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods. United States. doi:10.1007/S10958-018-4101-9.
Yankovskii, A. P. Tue . "Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods". United States. doi:10.1007/S10958-018-4101-9.
@article{osti_22773577,
title = {Analysis of the Spectral Stability of the Generalized Runge–Kutta Methods Applied to Initial-Boundary-Value Problems for Equations of the Parabolic Type. II. Implicit Methods},
author = {Yankovskii, A. P.},
abstractNote = {We consider specific realizations of different implicit generalized Runge–Kutta methods as applied to the numerical integration with respect to time of initial-boundary-value problems for the second-order parabolic equations and investigate their spectral stability. It is shown that all implicit generalized Runge–Kutta methods are unconditionally spectrally stable but some of them have the conditional property of monotonicity of the numerical solution with respect to time. The functions of spectral stability of the implicit generalized Runge–Kutta methods are rational. We compare the analytic solution of the nonstationary one-dimensional problem of heat conduction with the numerical solutions of this problem obtained by different implicit generalized Runge–Kutta methods. It is shown that, in this case, the application of the one-stage Radau methods with subsequent discretization of the problem with respect to the space variable leads to the classical forward finite difference scheme (Laasonen scheme), whereas the use of the one-stage Gauss–Legendre method leads to a six-point symmetric scheme (Crank–Nicolson scheme). It is shown that diagonally implicit generalized Nørsett and Burrage methods are realized in almost the same way as the one-stage Radau and Gauss–Legendre methods but their accuracy in the time step is 10–1000 times higher. On the basis of comparison of the numerical and analytic solutions, we conclude that, in order to get practically suitable numerical solutions without any restrictions on the time step, it is reasonable to use one- and three-stage generalized Radau methods or two- and four-stage Lobatto IIIC methods. All other explicit and implicit generalized Runge–Kutta methods require certain restrictions imposed on the time step.},
doi = {10.1007/S10958-018-4101-9},
journal = {Journal of Mathematical Sciences},
issn = {1072-3374},
number = 2,
volume = 236,
place = {United States},
year = {2019},
month = {1}
}