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Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows

Sergii Kivva

TL;DR

This work considers a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux, and applies the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations.

Abstract

First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented.

Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows

TL;DR

This work considers a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux, and applies the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations.

Abstract

First-order hydrostatic reconstruction (HR) schemes for shallow water equations are highly diffusive whereas high-order schemes can produce entropy-violating solutions. Our goal is to develop a flux correction with maximum antidiffusive fluxes to obtain entropy solutions of shallow water equations with variable bottom topography. For this purpose, we consider a hybrid explicit HR scheme whose flux is a convex combination of first-order Rusanov flux and high-order flux. The conditions under which the explicit first-order HR scheme for shallow water equations satisfies the fully discrete entropy inequality have been studied. The flux limiters for the hybrid scheme are calculated from a corresponding optimization problem. Constraints for the optimization problem consist of inequalities that are valid for the first-order HR scheme and applied to the hybrid scheme. We apply the discrete cell entropy inequality with the proper numerical entropy flux to single out a physically relevant solution to the shallow water equations. A nontrivial approximate solution of the optimization problem yields expressions to compute the required flux limiters. Numerical results of testing various HR schemes on different benchmarks are presented.
Paper Structure (13 sections, 6 theorems, 89 equations, 20 figures, 4 tables)

This paper contains 13 sections, 6 theorems, 89 equations, 20 figures, 4 tables.

Table of Contents

  1. Introduction
  2. First-Order Hydrostatic Reconstruction Scheme
  3. Cell Entropy Inequality for Fully Discrete HR Scheme
  4. Finding Flux Limiters
  5. Approximate Solution to the Optimization Problem
  6. Numerical Examples
  7. One-Dimensional Dam Break Over a Wet Flat Bed
  8. One-Dimensional Dam Break Over a Dry Bed
  9. Dam Break Over a Step.
  10. Steady Transcritical Flow With a Shock Over a Bump.
  11. Drainage on a Non-Flat Bottom
  12. 2D Partial Dam Break
  13. Conclusions

Key Result

Theorem 2.1

Assume that $c_{i+1/2} \geq \max \left( \vert u_i \vert ,\, \vert u_{i+1} \vert \right)$ for all $i$. Then for the following inequalities hold for the numerical solution of the system of equations eq:1.1-eq:1.2 where ${w^{\pm}_{i+1/2} = y^{\pm}_{i+1/2} +z_{i+1/2}}$.

Figures (20)

  • Figure 1: One-dimensional dam break over a wet flat bed. Comparisons of exact solutions with simulated water depths (left) and discharges (right) using HR1, ZL, and PP at t=10 s. The number of cells is N=100.
  • Figure 2: One-dimensional dam break over a wet flat bed. Comparisons of exact solutions with simulated water depths (left) and discharges (right) using HR2, LHE, and LHQE at t=10 s. The second row is a zoom in the area behind the shock. The number of cells is N=100.
  • Figure 3: One-dimensional dam break over a wet flat bed. Comparisons of numerical results obtained with FCT schemes whose flux limiters are computed using exact and approximate solutions to a linear programming problem with discrete entropy inequality and different constraints. The number of cells is N=100.
  • Figure 4: One-dimensional dam break over a dry bed. Comparisons of exact solutions with simulated water depths and discharges using HR1, ZL, and PP at time t=7 s. The number of cells is N=100.
  • Figure 5: One-dimensional dam break over a dry bed. Comparisons of exact solutions with simulated water depths and discharges using HR2, LHE, and LHQE at time t=7 s. The second row is a zoom in the area of the front of the moving water. The number of cells is N=100.
  • ...and 15 more figures

Theorems & Definitions (18)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Definition 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 8 more