Study of spectral stability of generalized Runge-Kutta methods in the initial-boundary problems for the wave equation

A general algorithm is developed to study the spectral stability of generalized multi-stage Runge-Kutta methods (RKMs) of different orders of accuracy for time integration of the wave equation. The stability function is obtained to estimate the spectral stability of these methods. Spectral stability of various explicit and implicit generalized RKMs is investigated. The behavior of the introduced stability function in a certain generalized RKM is found to be the same as in the previously studied case for the transport equation. It is shown that all the explicit generalized RKMs are spectral unstable, and the implicit generalized RKMs are spectral stable. Moreover, implicit methods based on the formulas of Rado, Lobatto IIIC, Merseta and Burridge possess the false attenuation property (asymptotic stability), and the methods of Gauss-Legendre, Lobatto IIIA, Lobatto IIIB of all orders of accuracy do not possess this property. Using the proposed stability functions, the spectral stability of Newmark’s family of methods is investigated. Computations demonstrate that one of the Newmark methods is a special case of the generalized RKM, namely, the one-step Gauss-Legendre (midpoint) method. Other Newmark methods are spectral unstable or they have the false attenuation property. A comparison is carried out between the approximate solutions obtained in terms of different generalized RKMs and Newmark methods and the exact solution of the problem of free vibrations of a string that is in equilibrium before its movement evoked by a concentrated force, immediately removed at the initial instant of time. The best finding in respect of a “simplicity of realization -achieved accuracy” ratio is the numerical result obtained from the three-stage fourth order diagonally implicit Burridge method, because the complexity of its realization is approximately the same as the Newmark methods, and its accuracy is higher by two orders of magnitude. It has been found that the algorithm developed to study the spectral stability of generalized RKMs and all theoretical results can be transferred with no changes to the parabolic equations, which contain the second time derivatives of unknown functions and describe the dynamic behavior of flexible beams or plates.