  The Rosenbrock-Krylov family of time integration schemes is an extension of
Rosenbrock-W methods that employs a specific Krylov based approximation of the
linear system solutions arising within each stage of the integrator. This work
proposes an extension of Rosenbrock-Krylov methods to address stability
questions which arise for methods making use of inexact linear system solution
strategies. Two approaches for improving the stability and efficiency of
Rosenbrock-Krylov methods are proposed, one through direct control of linear
system residuals and the second through a novel extension of the underlying
Krylov space to include stage right hand side vectors. Rosenbrock-Krylov
methods employing the new approaches show a substantial improvement in
computational efficiency relative to prior implementations.
