Contractivity-preserving implicit linear multistep methods

Abstract

Abstract.We investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on nonlinear and linear systems 4:tJ(t) = f(t, U(t)), where / satisfies a so-called circle condition in an arbitrary norm. The results for the two types of systems turn out to be closely related. We construct optimal multistep methods of given order and stepnumber, which allow the use of a maximal stepsize.

Monotonicity-Preserving Linear Multistep Methods

“…It was shown by Lenferink [13], in terms of contractivity for linear systems, that the threshold value K LM in (2.9) is bounded by 2 for all two-step methods of order p > 1. The optimal K LM = 2 is attained by the trapezoidal rule.…”

“…Related results for multistep methods were derived by Bolley and Crouzeix [2] in terms of positivity for linear systems. Contractivity for linear systems was studied by Spijker [17] and Lenferink [12,13]. The results in [2,4,13,17] also cover implicit methods; we discuss implicit methods in some detail in section 3.…”

“…In section 2 some related monotonicity properties are briefly discussed, together with existing results on multistep methods of the type (1.6) that were obtained in [2,4,12,13,15,17]. Then in section 3 we analyze the time-step restrictions for (1.2) of both explicit and implicit two-step methods with various starting procedures.…”

An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction

“…When using a time scheme with more stages than the order (s > p), the stability of the scheme is increased and the global computational time can decrease due to less re-computations. In this paper, we will only make reference to explicit discretization in time, as it is known [21,44,37,30] that the high-order implicit SSP schemes are not computationally efficient for very high-order because their CFL condition become at most twice the one of the forward Euler method.…”

Stability and boundedness in the numerical solution of initial value problems

“…In the context of solving just problems of type (1.4), there are still important LMMs for which no γ > 0 exists such that (1.12), (1.9) imply monotonicity, cf. [34] (p.283), [19], [20]. Moreover, the conditions on γ, given in the literature and relevant to (1.11), were obtained in the context of general (nonlinear) problems (1.1), and they are far from simple.…”