CONVERGENCE PROPERTIES OF THE RUNGE-KUTTA-CHEBYSHEV METHOD

The Runge-Kutta-Chebyshev method is an s-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length β proportional to s2. The method can be applied with s arbitrarily large, which is an attractive feature due to the proportionality of β with s2. The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for any s and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results. © 1990 Springer-Verlag.