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A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density

Jun She, Haiyun Dong, Maojun Li, Jianjun Ma

Abstract

In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method.

A well-balanced positivity-preserving discontinuous Galerkin method for shallow water models with variable density

Abstract

In this paper, we present a numerical scheme designed for coupled systems of variable-topography shallow water flow and solute transport. By integrating a variable-density system with an expression for relative density of mixtures, a novel formulation of the coupled system is derived. To ensure the well-balanced property, auxiliary variables are introduced to reformulate the variable-density shallow water equations into a new form, which is then discretized using the discontinuous Galerkin (DG) method with the Lax-Friedrichs (LF) flux as the numerical flux. By selecting appropriate values for the auxiliary variables, we demonstrate that the proposed method accurately preserves steady-state solutions under still water conditions, thereby verifying its well-balanced nature. Furthermore, sufficient conditions for preserving the positivity of both water depth and concentration are proposed and rigorously proven. A positivity-preserving limiter is introduced to enforce these conditions. Finally, a series of numerical examples are conducted to validate the computational accuracy and effectiveness of the proposed method.
Paper Structure (20 sections, 3 theorems, 53 equations, 8 figures, 2 tables)

This paper contains 20 sections, 3 theorems, 53 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Variable density shallow water system
  3. Fully discrete DG scheme
  4. Well-balanced DG reconstruction scheme
  5. Nontrivial steady state solutions
  6. Well-balanced DG scheme
  7. Positivity-Preserving DG scheme
  8. The cell averages of variables $h$ and $q_i$
  9. Sufficient conditions for positivity-preserving DG schemes
  10. Positivity-Preserving limiter
  11. Positivity-Preserving Well-Balanced DG Methods
  12. Numerical tests
  13. Small perturbation of a “still-water” state
  14. Parabolic Bowl
  15. Equilibrium preservation test for multi-component mixtures
  16. ...and 5 more sections

Key Result

Theorem 1

The proposed DG method defined in $(eq:19)$-$(eq:21)$ is a well-balanced scheme, meaning it preserves the “still-water” solution $(eq:15)$-$(eq:16)$.

Figures (8)

  • Figure 1: Two-dimensional views of the computed water surface elevation using the proposed schemes at times $t = 0.5, 0.9, 1.3$ and $1.7$.
  • Figure 2: Vertical cross-section of the water surface level $h + Z$ at $y = 0$ in a parabolic bowl at $t =T/8, T/4, 3T/8, T/2, 5T/8, 3T/4, 7T/8$ and $T$ (from left to right and from top to bottom). Circles numerical solution; dashed line exact water surface; solid line domain geometry.
  • Figure 3: Three-dimensional view of the numerical solution at time $t = 50$. Left: The computed water surface elevation and the bottom topography. Right: The computed volumetric concentration.
  • Figure 4: Cross sections at $y = 2.5$ of the computed volumetric concentrations for solutions with two constituents at time $t = 50$. Left: The computed water surface elevation and the bottom topography. Right: The computed volumetric concentration with two constituents.
  • Figure 5: Cross sections at $y = 2.5$ of the computed volumetric concentrations for solutions with four constituents at time $t = 50$. Left: The computed water surface elevation and the bottom topography. Right: The computed volumetric concentration with four constituents.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 1
  • Theorem 2