Using Diagonally Implicit Multistage Integration Methods for Solving Ordinary Differential Equations. Part 2: Implicit Methods.

Abstract

Abstract : New theoretical results for implicit methods of the recently invented and highly promising class of diagonally implicit multistage integration methods (DIMSIMs) are described. Some new A-stable and L-stable methods are derived, and conditions for a method to have an A-stable or L-stable FASAL (First Approximately Same As Last) implementation are derived. New alternative error estimates are proposed and successfully tested. The stability regions for alternative predictor-corrector DIMSIM implementations are derived. Implementation parameters for stiff solvers based on second and fifth order DIMSIMs are derived. Development of prototype stiff solvers of both second and fifth order stiff DIMSIM solvers is described. Results of successful tests on the Prothero-Robinson problem are reported and demonstrate that DIMSIMs may be used to develop efficient stiff solvers.