Explicit synchronous partitioned scheme for coupled reduced order models based on composite reduced bases

Abstract

This paper formulates, analyzes and demonstrates numerically a method for the explicit partitioned solution of coupled interface problems involving combinations of projection-based reduced order models (ROM) and/or full order models (FOMs). The method builds on the partitioned scheme developed in Peterson et al. (2019), which starts from a well-posed formulation of the coupled interface problem and uses its dual Schur complement to obtain an approximation of the interface flux. Explicit time integration of this problem decouples its subdomain equations and enables their independent solution on each subdomain. Extension of this partitioned scheme to coupled ROM–ROM or ROM–FOM problems requires formulations with non-singular Schur complements. To obtain these problems, we project a well-posed coupled FOM–FOM problem onto a composite reduced basis comprising separate sets of basis vectors for the interface and interior variables, and use the interface reduced basis as a Lagrange multiplier. Our analysis confirms that the resulting coupled ROM–ROM and ROM–FOM problems have provably non-singular Schur complements, independent of the mesh size and the reduced basis size. In the ROM–FOM case, analysis shows that one can also use the interface FOM space as a Lagrange multiplier. We illustrate the theoretical and computational properties of the partitioned scheme through reproductive and predictive tests for a model advection–diffusion transmission problem.