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On well-posed energy/entropy stable boundary conditions for the rotating shallow water equations

Kenneth Duru, Chuqiao Xu

TL;DR

This work addresses the challenge of designing well-posed, energy- and entropy-stable boundary conditions for the 2D rotating shallow water equations (RSWE) in vector invariant form under subcritical flows. It develops linear BCs based on Riemann invariants and Bernoulli potentials, and nonlinear BCs guided by linear consistency and stability, ensuring entropy stability for nonlinear IBVPs. The authors advance a high-order SBP-SAT discretization on curvilinear meshes that preserves the energy/entropy structure, and provide rigorous analysis alongside extensive numerical experiments to validate accuracy and boundary robustness. The results enable robust, nonreflecting regional simulations with complex geometries and Coriolis effects, with natural extensions to thermal RSWE and compressible Euler-type systems in future work.

Abstract

We derive and analyze well-posed, energy- and entropy-stable boundary conditions (BCs) for the two-dimensional linear and nonlinear rotating shallow water equations (RSWE) in vector invariant form. The focus of the study is on subcritical flows, which are commonly observed in atmospheric, oceanic, and geostrophic flow applications. We consider spatial domains with smooth boundaries and formulate both linear and nonlinear BCs using mass flux, Riemann's invariants, and Bernoulli's potential, ensuring that the resulting initial boundary value problem (IBVP) is provably entropy- and energy-stable. The linear analysis is comprehensive, providing sufficient conditions to establish the existence, uniqueness, and energy stability of solutions to the linear IBVP. For the nonlinear IBVP, which admits more general solutions, our goal is to develop nonlinear BCs that guarantee entropy stability. We introduce the concepts of linear consistency and linear stability for nonlinear IBVPs, demonstrating that if a nonlinear IBVP is both linearly consistent and linearly stable, then, for sufficiently regular initial and boundary data over a finite time interval, a unique smooth solution exists. Both the linear and nonlinear IBVPs can be efficiently solved using high-order accurate numerical methods. By employing high-order summation-by-parts operators to discretize spatial derivatives and implementing weak enforcement of BCs via penalty techniques, we develop provably energy- and entropy-stable numerical schemes on curvilinear meshes. Extensive numerical experiments are presented to verify the accuracy of the methods and to demonstrate the robustness of the proposed BCs and numerical schemes.

On well-posed energy/entropy stable boundary conditions for the rotating shallow water equations

TL;DR

This work addresses the challenge of designing well-posed, energy- and entropy-stable boundary conditions for the 2D rotating shallow water equations (RSWE) in vector invariant form under subcritical flows. It develops linear BCs based on Riemann invariants and Bernoulli potentials, and nonlinear BCs guided by linear consistency and stability, ensuring entropy stability for nonlinear IBVPs. The authors advance a high-order SBP-SAT discretization on curvilinear meshes that preserves the energy/entropy structure, and provide rigorous analysis alongside extensive numerical experiments to validate accuracy and boundary robustness. The results enable robust, nonreflecting regional simulations with complex geometries and Coriolis effects, with natural extensions to thermal RSWE and compressible Euler-type systems in future work.

Abstract

We derive and analyze well-posed, energy- and entropy-stable boundary conditions (BCs) for the two-dimensional linear and nonlinear rotating shallow water equations (RSWE) in vector invariant form. The focus of the study is on subcritical flows, which are commonly observed in atmospheric, oceanic, and geostrophic flow applications. We consider spatial domains with smooth boundaries and formulate both linear and nonlinear BCs using mass flux, Riemann's invariants, and Bernoulli's potential, ensuring that the resulting initial boundary value problem (IBVP) is provably entropy- and energy-stable. The linear analysis is comprehensive, providing sufficient conditions to establish the existence, uniqueness, and energy stability of solutions to the linear IBVP. For the nonlinear IBVP, which admits more general solutions, our goal is to develop nonlinear BCs that guarantee entropy stability. We introduce the concepts of linear consistency and linear stability for nonlinear IBVPs, demonstrating that if a nonlinear IBVP is both linearly consistent and linearly stable, then, for sufficiently regular initial and boundary data over a finite time interval, a unique smooth solution exists. Both the linear and nonlinear IBVPs can be efficiently solved using high-order accurate numerical methods. By employing high-order summation-by-parts operators to discretize spatial derivatives and implementing weak enforcement of BCs via penalty techniques, we develop provably energy- and entropy-stable numerical schemes on curvilinear meshes. Extensive numerical experiments are presented to verify the accuracy of the methods and to demonstrate the robustness of the proposed BCs and numerical schemes.
Paper Structure (23 sections, 12 theorems, 109 equations, 9 figures, 3 tables)

This paper contains 23 sections, 12 theorems, 109 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The nonlinear RSWE in vector invariant form
  3. Linear theory of IBVP for the RSWE
  4. Well-posedness of the linear RSWE IBVP
  5. Well-posed linear BCs
  6. The number of BCs for subcritical flows.
  7. Well-posed and stable BCs for subcritical flows.
  8. Nonlinear theory of IBVP for the RSWE
  9. Nonlinear stability and linear consistency
  10. Stable nonlinear BCs.
  11. Effects of the Coriolis force term on remote boundary data
  12. Numerical method and analysis
  13. Structure preserving curvilinear transformation
  14. Semi-discrete approximation
  15. Stable numerical boundary procedures
  16. ...and 8 more sections

Key Result

Lemma 1

Consider the linear differential operator $D$ given in eq6a-t subject to the BCs eq6c-t, $\mathcal{B}\mathbf{q} = 0$. Let $\mathrm{BT} = -\oint_{\partial \Omega}(gHh{u}_n + e{U}_n) dS$ be the boundary term given in eq:energy_conservation_linear and $W$ be the diagonal and positive definite weight ma

Figures (9)

  • Figure 1: Plots of the water height $h$ for the MMS solution \ref{['eqn:mms_lin']} at $t = 0$ for different mesh types
  • Figure 2: Convergence rate for 2D linear SWE with MMS for different mesh types
  • Figure 3: Convergence rate for 2D nonlinear SWE with MMS for different mesh types
  • Figure 4: The snapshots of the $y$-component of the particle velocity $\vb{v}$ for the linear RSWE on 2D rectangular geometry at $t = 5,10,15.$
  • Figure 5: The snapshots of the $y$-component of the particle velocity $\vb{v}$ for the linear RSWE on the panel of a cubesphere geometry at $t = 5,10,15.$
  • ...and 4 more figures

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Definition 4
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • ...and 21 more