  Systems driven by multiple physical processes are central to many areas of
science and engineering. Time discretization of multiphysics systems is
challenging, since different processes have different levels of stiffness and
characteristic time scales. The multimethod approach discretizes each physical
process with an appropriate numerical method; the methods are coupled
appropriately such that the overall solution has the desired accuracy and
stability properties. The authors developed the general-structure additive
Runge-Kutta (GARK) framework, which constructs multimethods based on
Runge-Kutta schemes.
  This paper constructs the new GARK-ROS/GARK-ROW families of multimethods
based on linearly implicit Rosenbrock/Rosenbrock-W schemes. For ordinary
differential equation models, we develop a general order condition theory for
linearly implicit methods with any number of partitions, using exact or
approximate Jacobians. We generalize the order condition theory to two-way
partitioned index-1 differential-algebraic equations. Applications of the
framework include decoupled linearly implicit, linearly implicit/explicit, and
linearly implicit/implicit methods. Practical GARK-ROS and GARK-ROW schemes of
order up to four are constructed.
