Model reduction of a coupled numerical model using proper orthogonal decomposition

Numerical models for variable-density flow and solute transport (VDFST) are widely used to simulate seawater intrusion and related problems. The mathematical model for VDFST is a coupled nonlinear dynamical system, so the numerical discretizations in time and space are usually required to be as fine as possible. As a result, fine-scale transient models require large computational time, which is a disadvantage for state estimation, forward prediction or model inversion. The purpose of this research is to develop mathematical and numerical methods to simulate VDFST via a model order reduction technique called Proper Orthogonal Decomposition (POD) designed for nonlinear dynamical systems. POD was applied to extract leading “model features” (basis functions) through singular value decomposition (SVD) from observational data or simulations (snapshots) of high-dimensional systems. These basis functions were then used in the Galerkin projection procedure that yielded low-dimensional (reduced-order) models. The original full numerical models were also discretized by the Galerkin Finite-Element Method (GFEM). The implementation of the POD reduced-order method was straightforward when applied to the full order model to the complex model. The developed GFEM-POD model was applied to solve two classic VDFST cases, the Henry problem and the Elder problem, in order to investigate the accuracy and efficiency of the POD model reduction method. Once the snapshots from full model results are obtained, the reduced-order model can reproduce the full model results with acceptable accuracy but with less computational cost in comparison with the full model, which is useful for model calibration and data assimilation problems. We found that the accuracy and efficiency of the POD reduced-order model is mainly determined by the optimal selection of snapshots and POD bases. Validation and verification experiments confirmed our POD model reduction procedure.