A high-order combined interpolation/finite element technique for evolutionary coupled groundwater-surface water problem

TL;DR

This paper introduces a high-order combined interpolation and finite element method for accurately and efficiently solving coupled groundwater-surface water flow problems in karst aquifers, with proven stability and convergence.

Contribution

It develops a novel high-order scheme that unconditionally stable, second-order in time, and of order d+1 in space, outperforming existing numerical methods for these coupled flow problems.

Findings

The method is unconditionally stable.

Achieves second-order temporal accuracy.

Converges with order d+1 in space.

Abstract

A high-order combined interpolation/finite element technique is developed for solving the coupled groundwater-surface water system that governs flows in karst aquifers. In the proposed high-order scheme we approximate the time derivative with piecewise polynomial interpolation of second-order and use the finite element discretization of piecewise polynomials of degree $d$ and $d+1$, where $d \geq 2$ is an integer, to approximate the space derivatives. The stability together with the error estimates of the constructed technique are established in $L^{\infty}(0,T;\text{\,}L^{2})$-norm. The analysis suggests that the developed computational technique is unconditionally stable, temporal second-order accurate and convergence in space of order $d+1$. Furthermore, the new approach is faster and more efficient than a broad range of numerical methods discussed in the literature for the given initial-boundary value problem. Some examples are carried out to confirm the theoretical results.