STIFFLY ACCURATE RUNGE–KUTTA METHODS FOR NONLINEAR EVOLUTION PROBLEMS GOVERNED BY A MONOTONE OPERATOR September 19, 2008

Abstract. Stiffly accurate implicit Runge–Kutta methods are studied for the time discretisation of nonlinear first-order evolution equations. The equation is supposed to be governed by a time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condi-tion. It is proven that the piecewise constant as well as the piecewise linear interpolant of the time-discrete solution converge towards the exact weak solu-tion, provided the Runge–Kutta method is consistent and satisfies a stability criterion that implies algebraic stability; examples are the Radau IIA and Lo-batto IIIC methods. The convergence analysis is also extended to problems involving a strongly continuous perturbation of the monotone main part. 1