  Many scientific applications require the solution of large initial-value
problems, such as those produced by the method of lines after
semi-discretization in space of partial differential equations. The
computational cost of implicit time discretizations is dominated by the
solution of nonlinear systems of equations at each time step. In order to
decrease this cost, the recently developed Rosenbrock-Krylov (ROK) time
integration methods extend the classical linearly-implicit Rosenbrock(-W)
methods, and make use of a Krylov subspace approximation to the Jacobian
computed via an Arnoldi process. Since the ROK order conditions rely on the
construction of a single Krylov space, no restarting of the Arnoldi process is
allowed, and the iterations quickly become expensive with increasing subspace
dimensions. This work extends the ROK framework to make use of the Lanczos
biorthogonalization procedure for constructing Jacobian approximations. The
resulting new family of methods is named biorthogonal ROK (BOROK). The Lanczos
procedure's short two-term recurrence allows BOROK methods to utilize larger
subspaces for the Jacobian approximation, resulting in increased numerical
stability of the time integration at a reduced computational cost. Adaptive
subspace size selection and basis extension procedures are also developed for
the new schemes. Numerical experiments show that for stiff problems, where a
large subspace used to approximate the Jacobian is required for stability, the
BOROK methods outperform the original ROK methods.
