Contributed Talk
Linearly Implicit General Linear Methods
Virginia Tech, USA
Abstract
Linearly implicit Runge–Kutta methods provide a fitting balance of implicit treatment of stiff systems and computational cost. In this paper we extend the class of linearly implicit Runge–Kutta methods to include multi-stage and multi-step methods. We provide the order condition theory 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 2 and type 4 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 linearly implicit Runge–Kutta methods. Our experiments show nominal order of convergence in test cases where Rosenbrock methods suffer from order reduction.
Joint work with Arash Sarshar, Virginia Tech, and Steven B. Roberts, Lawrence Livermore National Laboratory.