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

Eric Ngondiep, Areej A. Binsultan, Ibtisam M. Aldawish

TL;DR

The paper addresses simulating evolutionary coupling between groundwater and surface water flows by a Stokes–Darcy model with interface conditions. It introduces a hybrid higher-order interpolation/finite element scheme that uses a second-order time interpolation and space discretization of degree $d$ and $d+1$, achieving unconditional stability, temporal order $2$, and spatial convergence $O(h^{d+1})$. A rigorous stability and error analysis in $L^{\infty}(0,T;L^{2})$-type norms shows the method is unconditionally stable without stabilization terms, while numerical experiments confirm the predicted convergence behavior and computational efficiency. The approach offers a robust and fast tool for solving coupled groundwater-surface water systems and is proposed for extension to 3D nonlinear multi-physics problems such as elastodynamic sine-Gordon models.

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.

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

TL;DR

The paper addresses simulating evolutionary coupling between groundwater and surface water flows by a Stokes–Darcy model with interface conditions. It introduces a hybrid higher-order interpolation/finite element scheme that uses a second-order time interpolation and space discretization of degree and , achieving unconditional stability, temporal order , and spatial convergence . A rigorous stability and error analysis in -type norms shows the method is unconditionally stable without stabilization terms, while numerical experiments confirm the predicted convergence behavior and computational efficiency. The approach offers a robust and fast tool for solving coupled groundwater-surface water systems and is proposed for extension to 3D nonlinear multi-physics problems such as elastodynamic sine-Gordon models.

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 and , where is an integer, to approximate the space derivatives. The stability together with the error estimates of the constructed technique are established in -norm. The analysis suggests that the developed computational technique is unconditionally stable, temporal second-order accurate and convergence in space of order . 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.
Paper Structure (5 sections, 3 theorems, 144 equations, 4 figures)

This paper contains 5 sections, 3 theorems, 144 equations, 4 figures.

Table of Contents

  1. Introduction and preliminaries
  2. Construction of the higher-order computational approach
  3. Stability analysis and error estimates
  4. Numerical experiments
  5. General conclusion and future works

Key Result

Lemma 2.1

For any $w=(v,\phi),\text{\,\,}z=(u,\psi)\in V$, the bilinear operator $B(\cdot,\cdot)$ is symmetric and satisfies where $m=2$ or $3$, $\nu>0$ is the physical parameter given in equation $(1)$, $\widetilde{C}$ and $\widehat{C}$ are constants given in estimates $(38)$ and $(39)$, respectively, where $\lambda_{l}$ are the eigenvalues of the hydraulic tensor $\mathcal{K}$ defined in Section sec1. I

Figures (4)

  • Figure 1: Configuration of a coupled groundwater-surface water flow separated with an interface I
  • Figure 2: corresponding to Table 1 ($S_{0}=10^{-3}$ and $\eta=10^{-2}$)
  • Figure 3: associated with Table 2 ($S_{0}=10^{-7}$ and $\eta=10^{-2}$)
  • Figure 4: corresponding to Table 3 ($S_{0}=10^{-10}$ and $\eta=10^{-1}$)

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof