Unconditional Stability for Multistep ImEx Schemes: Theory
For researchers and practitioners using ImEx time-stepping schemes, this work provides a new class of methods with unconditional stability, addressing a key limitation in handling stiff splittings.
This paper introduces a new class of high-order linear ImEx multistep schemes that achieve large regions of unconditional stability, allowing time step selection based solely on accuracy. The schemes are particularly suited for problems where both implicit and explicit parts are stiff, such as variable-coefficient problems and incompressible Navier-Stokes equations.
This paper presents a new class of high order linear ImEx multistep schemes with large regions of unconditional stability. Unconditional stability is a desirable property of a time stepping scheme, as it allows the choice of time step solely based on accuracy considerations. Of particular interest are problems for which both the implicit and explicit parts of the ImEx splitting are stiff. Such splittings can arise, for example, in variable-coefficient problems, or the incompressible Navier-Stokes equations. To characterize the new ImEx schemes, an unconditional stability region is introduced, which plays a role analogous to that of the stability region in conventional multistep methods. Moreover, computable quantities (such as a numerical range) are provided that guarantee an unconditionally stable scheme for a proposed implicit-explicit matrix splitting. The new approach is illustrated with several examples. Coefficients of the new schemes up to fifth order are provided.