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Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu

TL;DR

The paper addresses the energy stability of the shallow water equations (SWEs) with a transmission boundary on open-sea boundaries. It develops an energy framework yielding the identity $\frac{d}{dt}E(t)=\sum_{i=1}^3 I_i(t;\Gamma)+I_4(t;\Omega)$, proving nonpositivity under Dirichlet/slip conditions and establishing a sufficient inequality for transmission boundaries when $0< c_0 \le \sqrt{2/\alpha}\,(1-\alpha)$ with $0<\alpha<1$. The authors validate the theory numerically using a finite difference scheme and a Lagrange-Galerkin (LG) scheme, showing the transmission boundary reduces artificial reflections and drives energy down in practice. The results provide both mathematical insight and practical confirmation for using transmission boundaries in tsunami/storm-surge simulations, while noting that a complete theoretical energy treatment of the transmission boundary remains a topic for future work.

Abstract

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmission boundary condition, an inequality for the energy estimate is proved under some assumptions to be satisfied in practical computation. Hence, it is recognized that the transmission boundary condition is reasonable in the sense that the inequality holds true. In the numerical part, based on the theoretical results, the energy estimate of the SWEs with a transmission boundary condition is confirmed numerically by a finite difference method (FDM). The choice of a positive constant c0 used in the transmission boundary condition is investigated additionally. Furthermore, we present numerical results by a Lagrange-Galerkin scheme, which are similar to those by the FDM. From the numerical results, it is found that the transmission boundary condition works well numerically.

Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition

TL;DR

The paper addresses the energy stability of the shallow water equations (SWEs) with a transmission boundary on open-sea boundaries. It develops an energy framework yielding the identity , proving nonpositivity under Dirichlet/slip conditions and establishing a sufficient inequality for transmission boundaries when with . The authors validate the theory numerically using a finite difference scheme and a Lagrange-Galerkin (LG) scheme, showing the transmission boundary reduces artificial reflections and drives energy down in practice. The results provide both mathematical insight and practical confirmation for using transmission boundaries in tsunami/storm-surge simulations, while noting that a complete theoretical energy treatment of the transmission boundary remains a topic for future work.

Abstract

Energy estimates of the shallow water equations (SWEs) with a transmission boundary condition are studied theoretically and numerically. In the theoretical part, using a suitable energy, we begin with deriving an equality which implies an energy estimate of the SWEs with the Dirichlet and the slip boundary conditions. For the SWEs with a transmission boundary condition, an inequality for the energy estimate is proved under some assumptions to be satisfied in practical computation. Hence, it is recognized that the transmission boundary condition is reasonable in the sense that the inequality holds true. In the numerical part, based on the theoretical results, the energy estimate of the SWEs with a transmission boundary condition is confirmed numerically by a finite difference method (FDM). The choice of a positive constant c0 used in the transmission boundary condition is investigated additionally. Furthermore, we present numerical results by a Lagrange-Galerkin scheme, which are similar to those by the FDM. From the numerical results, it is found that the transmission boundary condition works well numerically.

Paper Structure

This paper contains 10 sections, 4 theorems, 41 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Energy estimate
  4. Numerical results by a finite difference scheme
  5. A finite difference scheme
  6. Numerical results for five cases of boundary settings
  7. Numerical study of energy estimate
  8. Choice of $c_0$
  9. Numerical results by an LG scheme
  10. Conclusions

Key Result

Theorem 3.1

Suppose that a pair of smooth functions $(\phi, u): \overline{\varOmega} \times [0,T] \to \mathbb{R} \times \mathbb {R}^2$ satisfies prob. Then, we have

Figures (5)

  • Figure 1: The Bay of Bengal and the coastal region of Bangladesh
  • Figure 2: Model domain
  • Figure 3: Color contours of $\eta_h^k$ by finite difference scheme \ref{['scheme']} for the five cases $(i)$-$(v)$ discussed in Subsection \ref{['subsec4.2']}.
  • Figure 4: Graphs of $E_h^k$ (left), $\sum_{i=1}^4 I_{hi}^k \approx \frac{d}{dt} E(t)$ (center) and $I_{hi}^k$, $i=1, \ldots, 4$, (right) versus $t=t^k~(\ge 0, k\in\mathbb{Z})$ for the five cases $(i)$-$(v)$.
  • Figure 5: Color contours of $\eta_h^k$ by Lagrange--Galerkin scheme \ref{['scheme:LG']} for the five cases $(i)$-$(v)$ discussed in Section \ref{['sec5']}.

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['th1']}
  • Corollary 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Remark 3.6