A reduced order modeling approach with Petrov–Galerkin projection based on hybrid snapshot simulation in Semilinear-PDEs

Abstract

This paper proposes an efficient and economic framework to generate reduced order models (ROMs) with Petrov–Galerkin projection based on hybrid snapshot simulation to extraordinarily enhance the speedup and enlighten the representation of snapshot data in Semilinear Partial Differential Equations (Semilinear-PDEs). The total snapshot simulation is divided into multiple temporal intervals with the alternations in each interval between two types of models: the full order model (FOM) and the local ROM based on Petrov–Galerkin projection (ROM-PG). During the snapshot training process, the simulation can determine the model type on-the-fly by the switch criteria introduced in Bai and Wang (2022) and continuously update the subspace by incremental singular value decomposition (iSVD) to preserve the subspace representation. In numerical experiments, we consider two typical Semilinear-PDEs to illustrate the proposed method successfully in finite volume element discretization: Semilinear-Parabolic Equation (Semilinear-Parabolics) and Semilinear-Convection–Diffusion Equation (Semilinear-ConvDiffs). It is found that the subspace updating is directly related to the POD numbers, the parameter μ (μp in Semilinear-Parabolics or μc in Semilinear-ConvDiffs). The tolerance of criteria also significantly affects the performance of computational acceleration. The quantitative analyses of the criteria and the brief discussion of stability about the framework are also performed to clarify the model switch and stability condition, respectively, due to no detailed discussion in the previous work (Bai and Wang 2022).