A Newton - Galerkin Finite Element Method to solve reduced dimensional Variable Density Flow and Solute Transport Equations using Proper Orthogonal Decomposition method

Abstract

The mathematical model of Variable Density Flow and Solute Transport (VDFST) is a time dependent, coupled and nonlinear dynamical system that is widely used to simulate seawater intrusion and related problems. The numerical problem is relatively easy to solve when the transport of solute does not affect fluid density, but when there are big differences in density, the problem of solute transport is much more difficult to solve because of the high degree of nonlinearity. The numerical discretizations of VDFST in time and space are usually required to be as fine as possible, but due to their high dimensional structure there is a strong need to reduce the computational costs and storage requirements. Proper orthogonal decomposition (POD), as a model order reduction (MOR) technique, aims to lower the computational complexity by approximating the large-scaled discretized state equations using a low-dimensional model. POD is an effective numerical technique to reduce the computational cost for state estimation, forward prediction and inverse modeling. In this research, POD was used with the Galerkin finite element method (GFEM) and the Newton iteration approach to reduce the computational time and the relative error between the results we obtain from the reduced dimensional and high dimensional models of VDFST. The modified Henry problem and Elder problem were used to demonstrate the capability of the model. It was showed that the reduced dimensional model that was solved with Newton iteration approach can reproduce and predict the full model results very accurately with much less computational time in comparison with the full dimensional model and the reduced dimensional model that was solved with coupling iteration approach.