A unified computational framework for developing general multi-time-step (MTS) coupling schemes for time-dependent problems

Abstract Multi-time-step (MTS) methods solve temporal multi-scale problems by decomposing the spatial problem domain into multiple subdomains and integrating each subdomain with a time step that is suitable for capturing its response. A critical limitation of existing MTS methods, however, is that they are based on enforcing continuity of velocities across the interface between subdomains and consequently lead to erroneous drifts in displacements that show up as nonphysical gaps and overlaps in the solution. In this article, we propose a new MTS framework that enables simultaneous enforcement of multiple continuity constraints between subdomains and the use of general interpolation schemes for associated Lagrange multipliers. To make the framework as general as possible, we first express almost all existing time integration schemes in a Generalized Runge–Kutta–Nyström (GRKN) form, which then serve as the underlying time integration schemes for individual subdomains. MTS methods designed using the proposed framework are able to achieve unconditional stability by introducing a scalar auxiliary variable (SAV) based on physical energy of the system and ensuring that this energy remains bounded during the simulation. We design new MTS methods based on piecewise-linear and quadratic interpolations of Lagrange multipliers with simultaneous enforcement of velocity and displacement continuity (VCDC) to demonstrate typical use-cases of the proposed framework. We present several numerical examples to show that solutions obtained using the new VCDC MTS methods are able to achieve greater accuracy for the same computational cost in comparison to single time integration schemes and existing MTS methods.