A novel space/time integration technology via altogether different space and time stepping methods for nonlinear first-order systems

Purpose The purpose of this paper is to describe a novel implementation of a multispatial method, multitime-scheme subdomain differential algebraic equation (DAE) framework allowing a mix of different space discretization methods and different time schemes by a robust generalized single step single solve (GS4) family of linear multistep (LMS) algorithms on a single body analysis for the first-order nonlinear transient systems. Design/methodology/approach This proposed method allows the coupling of different numerical methods, such as the finite element method and particle methods, and different implicit and/or explicit algorithms in each subdomain into a single analysis with the GS4 framework. The DAE, which constrains both space and time in multi-subdomain analysis, combined with the GS4 framework ensures the second-order time accuracy in all primary variables and Lagrange multiplier. With the appropriate GS4 parameters, the algorithmic temperature rate variable shift can be matched for all time steps using the DAE. The proposed method is used to solve various combinations of spatial methods and time schemes between subdomains in a single analysis of nonlinear first-order system problems. Findings The proposed method is capable of coupling different spatial methods for multiple subdomains and different implicit/explicit time integration schemes in the GS4 framework while sustaining second-order time accuracy. Originality/value Traditional approaches do not permit such robust and flexible coupling features. The proposed framework encompasses most of the LMS methods that are second-order time accurate and unconditionally stable.