The solution of initial value problems is a fundamental component of many scientific simulations of physical phenomena. In many cases these initial value problems arise from a method of lines approach to solving partial differential equations, resulting in very large systems of equations that require the use of numerical time integration methods to solve. Many problems of scientific interest exhibit stiff behavior for which implicit methods are favorable, however standard implicit methods are computationally expensive. They require the solution of one or more large nonlinear systems at each timestep, which can be impractical to solve exactly and can behave poorly when solved approximately. The recently introduced ``lightly-implicit'' K-methods seek to avoid this issue by directly coupling the time integration methods with a Krylov based approximation of linear system solutions, treating a portion of the problem implicitly and the remainder explicitly.

This work seeks to further two primary objectives: evaluation of these K-methods in large-scale parallel applications, and development of new linearly implicit methods for contexts where improvements can be made. To this end, Rosenbrock-Krylov methods, the first K-methods, are examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock-Krylov methods, and linearly implicit multistep methods.

For the scalability evaluation of Rosenbrock-Krylov methods, two parallel contexts are considered: a GPU accelerated model and a distributed MPI parallel model. In both cases, the most significant performance bottleneck is the need for many vector dot products, which require costly parallel reduce operations.

Biorthogonal Rosenbrock-Krylov methods are an extension of the original Rosenbrock-Krylov methods which replace the Arnoldi iteration used to produce the Krylov approximation with Lanczos biorthogonalization, which requires fewer vector dot products, leading to lower overall cost for stiff problems.

Linearly implicit multistep methods are a new family of implicit multistep methods that require only a single linear solve per timestep; the family includes W- and K-method variants, which admit arbitrary or Krylov based approximations of the problem Jacobian while maintaining the order of accuracy. This property allows for a wide range of implementation optimizations.

Finally, all the new methods proposed herein are implemented efficiently in the MATLODE package, a Matlab ODE solver and sensitivity analysis toolbox, to make them available to the community at large.