Stability of Implicit-Explicit Linear Multistep Methods
Abstract In many applications, such as atmospheric chemistry, large systems of ordinary differential equations (ODEs) with both stiff and nonstiff parts have to be solved numerically. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intended for such applications. The paper focuses on the linear stability of popular second order methods like extrapolated BDF, Crank-Nicolson leap-frog and a particular class of Adams methods. We present results for problems with decoupled eigenvalues and comment on some specific CFL restrictions associated with advection terms.
The specification of non-functional requirements, e. g., on timing forms an essential part of modern system design. Modeling languages such as MARTE/CCSL provide dedicated description means enabling engineers to formally define the ticking of the clocks to be implemented in terms of clock constraints and the actually intended timing behavior in terms of instant relations. But thus far, instant relations have only been utilized in order to monitor the correct execution of the clock constraints. In this work, we propose a methodology which, for the first time, verifies clock constraints against the given instant relations. To this end, the timing behavior is represented in terms of an automaton followed by its verification through satisfiability solvers. A case study illustrates the application of the proposed methodology.