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Asymptotic Analysis of Shallow Water Moment Equations

Mieke Daemen, Julio Careaga, Zhenning Cai, Julian Koellermeier

Abstract

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.

Asymptotic Analysis of Shallow Water Moment Equations

Abstract

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.
Paper Structure (35 sections, 5 theorems, 96 equations, 10 figures, 9 tables)

This paper contains 35 sections, 5 theorems, 96 equations, 10 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Shallow Water Moment Equations
  3. Shallow water model
  4. Deriving SWME from the Navier–Stokes equations
  5. Vertical coordinate mapping
  6. Polynomial expansion
  7. Closed-form SWME
  8. Asymptotic approximations of the SWME with viscous slip scaling
  9. Asymptotic expansion for SWME with $N=1$
  10. The $\mathcal{O}(\varepsilon^{-1})$-approximation of the SWME1
  11. The $\mathcal{O}(\varepsilon^{0})$-approximation of the SWME1
  12. The $\mathcal{O}(\varepsilon^{1})$-approximation of the SWME1
  13. The closed reduced SWME1 model
  14. Hyperbolicity of the RSWME1
  15. Asymptotic expansion for SWME with $N=2$
  16. ...and 20 more sections

Key Result

Theorem 1

The RSWME1 system eq: ASWME1 admits a globally hyperbolic regularisation by adding the following term to the left-hand side of its last equation:

Figures (10)

  • Figure 1: Wave with a sharp gradient test comparing the SWE (red, solid), RSWME1 (green, dashed), and SWME1 (blue, dotted) models at $t=2.0$. The results are presented for the height $h$ (top row) and velocity $u_m$ (bottom row) across three values of $\varepsilon$: $\varepsilon = 0.01$ (left), $\varepsilon = 0.1$ (middle), and $\varepsilon = 1$ (right).
  • Figure 2: Wave with a sharp gradient test comparing the SWE (red, solid), RSWME1 (green, dashed), and SWME1 (blue, dotted) models at $t=2.0$. The results are presented for the moment coefficient $\alpha_1$ across three values of $\varepsilon$: $\varepsilon = 0.01$ (left), $\varepsilon = 0.1$ (middle), and $\varepsilon = 1$ (right).
  • Figure 3: Wave with a sharp gradient test comparing the SWE (red, solid), RSWME1 (green, dashed), and SWME1 (blue, dotted) models at $t=2.0$. The post processed $\partial_x(h^4)$-value computed with Godunov reconstruction is shown across three values of $\varepsilon$: $\varepsilon = 0.01$ (left), $\varepsilon = 0.1$ (middle), and $\varepsilon = 1$ (right).
  • Figure 4: Smooth sine-wave test comparing the SWE (red, solid), RSWME1 (green, dashed), and SWME1 (blue, dotted) models at $t=2.0$. The results are presented for the variables height $h$ (top row) and average velocity $u_m$ (bottom row) across three values of $\varepsilon$: $\varepsilon = 0.01$ (left), $\varepsilon = 0.1$ (middle), and $\varepsilon = 1$ (right).
  • Figure 5: Smooth sine-wave test comparing the SWE (red, solid), RSWME1 (green, dashed), and SWME1 (blue, dotted) models at $t=2.0$. The results are presented for the moment coefficient $\alpha_1$ across three values of $\varepsilon$: $\varepsilon = 0.01$ (left), $\varepsilon = 0.1$ (middle), and $\varepsilon = 1$ (right).
  • ...and 5 more figures

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1: Hyperbolic regularisation of RSWME1
  • proof
  • Theorem 2: Hyperbolic regularisation of RSWME2
  • proof
  • Theorem 3: Identical closed RSWME systems
  • proof
  • Lemma 1
  • proof
  • Theorem 3: Identical closed RSWME systems
  • ...and 1 more