Study of the Spectral Stability of Generalized Runge–Kutta Methods in the Initial Problem for the Transfer Equation

Abstract

We study the problem of spectral stability of generalized Runge–Kutta methods of various orders of accuracy as applied to the numerical integration of the initial-value problem for the transfer equation and compare the approximate solutions obtained by using various generalized Runge–Kutta methods with the exact solution for complex oscillating initial conditions with derivatives large in the absolute value. It is shown that some classical finite-difference schemes of integration of the initial-boundary-value problem for the transfer equation are obtained as a result of successive application of generalized and ordinary Runge–Kutta methods with respect to all independent variables.