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Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system

Bernard Di Martino, Chourouk El Hassanieh, Edwige Godlewski, Julien Guillod, Jacques Sainte-Marie

TL;DR

The paper advances the mathematical analysis of hydrostatic free-surface Euler flows by recasting the problem through a semi-Lagrangian change of variables, producing a quasi-linear system with horizontal derivatives and a vertical integral operator. It rigorously analyzes the operator spectrum to establish generalized hyperbolicity, derives Riemann invariants, and constructs explicit stationary and shallow-water solutions to illuminate the model's behavior. A key novelty is an exact multilayer $\mathbb{P}_0$ discretization that preserves the transformed system and yields a tractable spectrum analysis, with convergence results showing the discrete spectrum tends to the continuous spectrum under reasonable regularity and monotonicity assumptions. The findings have practical implications for stable numerical approximations of geophysical flows with free surfaces, providing conditions under which real eigenvalues govern stability and offering a framework to assess discretizations via spectral properties.

Abstract

By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.

Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system

TL;DR

The paper advances the mathematical analysis of hydrostatic free-surface Euler flows by recasting the problem through a semi-Lagrangian change of variables, producing a quasi-linear system with horizontal derivatives and a vertical integral operator. It rigorously analyzes the operator spectrum to establish generalized hyperbolicity, derives Riemann invariants, and constructs explicit stationary and shallow-water solutions to illuminate the model's behavior. A key novelty is an exact multilayer discretization that preserves the transformed system and yields a tractable spectrum analysis, with convergence results showing the discrete spectrum tends to the continuous spectrum under reasonable regularity and monotonicity assumptions. The findings have practical implications for stable numerical approximations of geophysical flows with free surfaces, providing conditions under which real eigenvalues govern stability and offering a framework to assess discretizations via spectral properties.

Abstract

By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer -discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.
Paper Structure (20 sections, 25 theorems, 168 equations, 9 figures)

This paper contains 20 sections, 25 theorems, 168 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Semi-Lagrangian formulation of hydrostatic Euler system
  3. Free-surface hydrostatic Euler system
  4. Derivation of the semi-Lagrangian formulation
  5. Proofs of the equivalence of the two formulations
  6. Particular solutions
  7. Stationary solutions of hydrostatic Euler system depending only on z
  8. Stationary solutions of the semi-Lagrangian formulation
  9. Shallow water flows
  10. A flow with an horizontal velocity depending on the vertical coordinate
  11. Spectrum and Riemann invariants
  12. Characterization of the spectrum
  13. Limiting cases
  14. Generalized Riemann invariants
  15. Case with variable topography
  16. ...and 5 more sections

Key Result

Theorem 2.3

Let $s\geq1$, $T>0$, $p^{a}=0$, and $z_b\in C_{b}^{s}(\mathbb{R}^{d})$. If $(\eta,\boldsymbol{u},w)$ is a solution of eq:euler_3d such that $\eta\in C_{b}^{s}((0,T)\times\mathbb{R}^{d})$, $\boldsymbol{u}\in C_{b}^{s}((0,T)\times\Omega_{t})^d$, and $w\in C_{b}^{s}((0,T)\times\Omega_{t})$, then for $\ is an orientation preserving $C^{s}$-diffeomorphism, provided it is a diffeomorphism at $t=0$, i.e.

Figures (9)

  • Figure 1: The three dimensional set-up for the hydrostatic Euler system with free-surface, where $\boldsymbol{x}=(x,y)$ and $\boldsymbol{u}=(u,v)$.
  • Figure 2: Representation of the explicit steady solution given by \ref{['cor:sol_anal_sta']} for $F(\lambda)=1+\lambda$ and $G(x)=\frac{7}{10}\sin(x) + \frac{2}{10}\tanh(5x)$. The line $\lambda=0$ corresponds to the topography $z_b$, the line $\lambda=1$ to the free surface $\eta$, whereas the intermediate dotted lines correspond to values of $\lambda$ in between.
  • Figure 3: Explicit flow with vorticity proposed in \ref{['subsec:flow_vorticity']} with $\eta_0=1$.
  • Figure 4: Graph of $\phi$ defined by \ref{['eq:example_vorticity_phi']} for $\eta_0=1$ at times $t=0,2,4,6$.
  • Figure 5: Representation of the various conditions on $c$ used in the proof of \ref{['thm:spectrum']}.
  • ...and 4 more figures

Theorems & Definitions (64)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.9
  • proof
  • ...and 54 more