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Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

Shihua Gong, Young-Ju Lee, Yukun Li, Yue Yu

TL;DR

The paper addresses preserving positivity and maximum principles in DG discretizations for coupled flow and transport. It introduces a locally conservative flux notion and ties it to the Dawson–Sun–Wheeler compatibility framework, establishing LR-DG and BMS-DG equivalence and linking to streamline-based characteristics. Under a locally conservative flux of degree $k$, the transport solution attains $L^2$ stability for $k \ge 2k_c$ and zeroth-order accuracy for $k \ge k_c$, with $k_c=0$ yielding positivity and a discrete maximum principle for piecewise-constant transport. Numerical experiments in 1D and 2D corroborate the theory and demonstrate the critical role of local conservation in preventing nonphysical oscillations and ensuring mass conservation in flow–transport simulations.

Abstract

We introduce a new concept of the locally conservative flux and investigate its relationship with the compatible discretization pioneered by Dawson, Sun and Wheeler [11]. We then demonstrate how the new concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Suli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have also been performed to justify our theoretical findings

Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

TL;DR

The paper addresses preserving positivity and maximum principles in DG discretizations for coupled flow and transport. It introduces a locally conservative flux notion and ties it to the Dawson–Sun–Wheeler compatibility framework, establishing LR-DG and BMS-DG equivalence and linking to streamline-based characteristics. Under a locally conservative flux of degree , the transport solution attains stability for and zeroth-order accuracy for , with yielding positivity and a discrete maximum principle for piecewise-constant transport. Numerical experiments in 1D and 2D corroborate the theory and demonstrate the critical role of local conservation in preventing nonphysical oscillations and ensuring mass conservation in flow–transport simulations.

Abstract

We introduce a new concept of the locally conservative flux and investigate its relationship with the compatible discretization pioneered by Dawson, Sun and Wheeler [11]. We then demonstrate how the new concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Suli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have also been performed to justify our theoretical findings
Paper Structure (16 sections, 9 theorems, 79 equations, 4 figures, 2 tables)

This paper contains 16 sections, 9 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Continuous Maximum Principle
  4. Discontinuous Galerkin Finite Element Formulation for flow and transport
  5. DG formulation for the flow and the concept of the local conservation
  6. DG formulation for the transport
  7. Compatibility condition and its relationship to the local conservation
  8. $L^2$-Stability, Positivity and Discrete maximum principle (DMP)
  9. The Lesaint-Raviart DG method and its equivalence with the BMS-DG scheme
  10. The relation between the Lesaint-Raviart DG method and the Characteristic method
  11. Numerical Experiments
  12. 1D case of propagating a concentration front through the domain
  13. 2D case with no external force and no diffusion
  14. 2D case with external force for injection and extraction
  15. 2D case to study the error accumulation for concentration
  16. ...and 1 more sections

Key Result

Theorem 2.1

\newlabelthm_wmp0 For the initial condition $c^0$, the boundary condition $c_{\rm I}$ and specified source $\widetilde{c}$, we let $M=\max\{|c^0|,|c_{\rm I}|,|\widetilde{c}|\}$. Under the condition of the strong conservation eqn:conservation_mass, we can show that the solution to equations eqn:ful

Figures (4)

  • Figure 1: Concentration values at $T=0.5$ with piecewise constant function(left) and piecewise linear function(middle); $L^2$ norm for each time step using $DG_2$ for the transport equation and $DG_1$, $DG_2$, $DG_3$ and $DG_4$ for the flow equation.(right).
  • Figure 2: Left: Domain; Middle: K-block; Right: Pressure.
  • Figure 3: Concentration values at $T=0.5, 1.0, 1.5, 3.0$ (from left to right).
  • Figure 4: Concentration values at $T=0, 4, 6, 10$ with injection and extraction. The first row used the piecewise constant function and the second row used the piecewise linear function.

Theorems & Definitions (17)

  • Theorem 2.1
  • Proof 1
  • Definition 3.1
  • Theorem 3.2
  • Proof 2
  • Lemma 4.1
  • Proof 3
  • Lemma 4.2
  • Proof 4
  • Theorem 4.3
  • ...and 7 more