Generalization of the Runge–Kutta methods and their application tointegration of initial-boundary value problems of mathematical physics

An idea is proposed and tested to generalize the Runge–Kutta methods to a bidimensional case for the approximate integration of the initial-boundary value problems corresponding to the partial differential equations. It is shown that some classical finite difference schemes of integration of the equation of transport and non-stationary one-dimensional heat conductivity can be obtained as consequence of such generalization. New schemes of high orders of accuracy for various problems of mathematical physics are obtained. Stability of these schemes is proved, and results of calculations for problems with large gradients of the solution are presented. On concrete examples it is shown that classical schemes of low orders of accuracy unsatisfactorily describe solutions of such problems, and the schemes of high orders constructed by means of the generalized Runge–Kutta methods presented, give a good approximation to exact solutions.