Study of the spectral stability of generalized Runge-Kutta methods applied to numerical integration of the initial-bounary value problem for the transport equation

Based on the harmonics method, the spectral stability of the generalized Runge-Kutta methods of the first and second order of accuracy in time step is investigated analytically for numerical integration of the initial-boundary value problem for the transport equation. It is shown that some classical explicit and implicit finite-difference schemes of integration of the initial-boundary value problem for the transport equation is a consequence of the consistent application of the generalized and ordinary Runge-Kutta methods to all independent variables. A general algorithm for the analysis is developed to study the spectral stability of the generalized multi-stage Runge-Kutta methods of different orders of accuracy for transport equation integration. The spectral stability of various explicit and implicit generalized Runge-Kutta methods is explored. It has been found that all the explicit methods are spectral unstable, and all the implicit methods are spectral stable, and the implicit methods based on the formulas of Rado, Lobatto IIIC, Nereta and Burridge are asymptotically stable, whereas the methods of Gauss-Legendre, Lobatto IIIA, Lobatto IIIB of all orders of accuracy (although they are spectral stable) are not asymptotically stable. Comparison of the approximate solutions obtained by different generalized Runge-Kutta methods with the exact solution is carried out under complex-fluctuating initial conditions involving large modulo derivatives conditionally modeling the impact of shocks. It turns out that the numerical results obtained by the Rado formulas of high order of accuracy are the best. The possibilities of using the proposed approach for studying the spectral stability of the generalized Runge-Kutta methods applied to numerical integration of systems of first-order equations of hyperbolic type in one- and multi-dimensional cases are outlined.