Linearly-Implicit General Linear Methods

Published in Preprint, 2021

Linearly implicit Runge-Kutta methods provide a fitting balance between implicit treatment of stiff systems and computational costs. We extend the class of linearly implicit Runge-Kutta methods to include multi-stage and multi-step methods. We discuss the order condition to achieve high stage order and overall accuracy while admitting arbitrary Jacobians. Several classes of Linearly implicit general linear methods (GLMs) are discussed based on existing families such as type-II and Type-IV GLMs, two-step Runge-Kutta methods, Parallel IMEX GLMs, and BDF methods. We investigate the stability implications for stiff problems and provide numerical studies for the behavior of our methods compared to others.