Before revision:  The new method is demonstrated to be effective on various high-order finite-difference and finite-element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th order accuracy.
After revision:  The new method can be used with arbitrary existing preconditioners for backward Euler-type time stepping schemes, and is amenable to the use of three-term recursion Krylov methods when the underlying spatial discretization is symmetric. The new method is demonstrated to be effective on various high-order finite-difference and finite-element discretizations of linear parabolic and hyperbolic problems, demonstrating fast, scalable solution of up to 10th order accuracy.