Second-order Rosenbrock-Exponential (ROSEXP) Methods for Partitioned Differential Equations

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the accuracy and stability of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, 2 nd order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. In addition to comparing the new integrators to previously proposed schemes, our numerical experiments validate the convergence and efficiency of the new methods.