  This paper develops a new class of Rosenbrock-type integrators based on a
Krylov space solution of the linear systems. The new family, called
Rosenbrock-Krylov (Rosenbrock-K), is well suited for solving large scale
systems of ODEs or semi-discrete PDEs. The time discretization and the Krylov
space approximation are treated as a single computational process, and the
Krylov space properties are an integral part of the new Rosenbrock-K order
condition theory developed herein. Consequently, Rosenbrock-K methods require a
small number of basis vectors determined solely by the temporal order of
accuracy. The subspace size is independent of the ODE under consideration, and
there is no need to monitor the errors in linear system solutions at each
stage. Numerical results show favorable properties of Rosenbrock-K methods when
compared to current Rosenbrock and Rosenbrock-W schemes.
