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

Journal Article · Tue Jan 15 04:00:00 EST 2019 · Journal of Mathematical Sciences

DOI:https://doi.org/10.1007/S10958-018-4101-9· OSTI ID:22773577

Yankovskii, A. P. ^[1]

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1. Khristianovich Institute of Theoretical and Applied Mechanics of Siberian Branch of RAS (Russian Federation)

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.

• OSTI ID: 22773577
• Journal Information: Journal of Mathematical Sciences, Journal Name: Journal of Mathematical Sciences Journal Issue: 2 Vol. 236; ISSN JMTSEW; ISSN 1072-3374
• Country of Publication: United States
• Language: English

Related Subjects

42 ENGINEERING
97 MATHEMATICS AND COMPUTING
ACCURACY
ANALYTICAL SOLUTION
BOUNDARY-VALUE PROBLEMS
COMPARATIVE EVALUATIONS
EQUATIONS
LEGENDRE POLYNOMIALS
ONE-DIMENSIONAL CALCULATIONS
RUNGE-KUTTA METHOD
STABILITY
SYMMETRY
THERMAL CONDUCTION