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Well-balanced fifth-order finite volume WENO schemes with constant subtraction technique for shallow water equations

Lidan Zhao, Zhanjing Tao, Min Zhang

Abstract

In this paper, we propose a new well-balanced fifth-order finite volume WENO method for solving one- and two-dimensional shallow water equations with bottom topography. The well-balanced property is crucial to the ability of a scheme to simulate perturbation waves over the ``lake-at-rest'' steady state such as waves on a lake or tsunami waves in the deep ocean. We adopt the constant subtraction technique such that both the flux gradient and source term in the new pre-balanced form vanish at the lake-at-rest steady state, while the well-balanced WENO method by Xing and Shu [Commun. Comput. Phys., 2006] uses high-order accurate numerical discretization of the source term and makes the exact balance between the source term and the flux gradient, to achieve the well-balanced property. The scaling positivity-preserving limiter is used for the water height near the dry areas. The fifth-order WENO-AO reconstruction is used to construct the solution since it has better resolution than the WENO-ZQ and WENO-MR reconstructions for the perturbation of steady state flows. Extensive one- and two-dimensional numerical examples are presented to demonstrate the well-balanced, fifth-order accuracy, non-oscillatory, and positivity-preserving properties of the proposed method.

Well-balanced fifth-order finite volume WENO schemes with constant subtraction technique for shallow water equations

Abstract

In this paper, we propose a new well-balanced fifth-order finite volume WENO method for solving one- and two-dimensional shallow water equations with bottom topography. The well-balanced property is crucial to the ability of a scheme to simulate perturbation waves over the ``lake-at-rest'' steady state such as waves on a lake or tsunami waves in the deep ocean. We adopt the constant subtraction technique such that both the flux gradient and source term in the new pre-balanced form vanish at the lake-at-rest steady state, while the well-balanced WENO method by Xing and Shu [Commun. Comput. Phys., 2006] uses high-order accurate numerical discretization of the source term and makes the exact balance between the source term and the flux gradient, to achieve the well-balanced property. The scaling positivity-preserving limiter is used for the water height near the dry areas. The fifth-order WENO-AO reconstruction is used to construct the solution since it has better resolution than the WENO-ZQ and WENO-MR reconstructions for the perturbation of steady state flows. Extensive one- and two-dimensional numerical examples are presented to demonstrate the well-balanced, fifth-order accuracy, non-oscillatory, and positivity-preserving properties of the proposed method.

Paper Structure

This paper contains 7 sections, 5 theorems, 88 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Well-balanced WENOAO-CST method for 1D SWEs
  3. The calculation of $\mathcal{L}(\overline{\mathbf{U}}_i)$
  4. Preservation of positivity for water height $h$
  5. Well-balanced WENOAO-CST method for 2D SWEs
  6. Numerical examples
  7. Conclusions

Key Result

Proposition 2.1

The semi-discrete finite volume WENOAO-CST scheme (semi-discrete-wb) for 1D SWEs is well-balanced, i.e., it preserves the lake-at-rest steady state $H=h+b = C,~hu=0$.

Figures (17)

  • Figure 1: The quadrature points in two dimensions.
  • Figure 2: The $L^1$ and $L^\infty$ errors vs. CPU time of the water height $h$ and discharge $hu$ by the WENOAO-CST and WENOJS-XS methods. $T= 0.1$.
  • Figure 3: The water height $h$ and velocity $u$ for the tidal wave flow test. $T=7,552.13$ and $N_x=200$. Square: WENOAO-CST; plus: WENOJS-XS; solid line: exact solutions.
  • Figure 4: The surface level $h + b$ and discharge $hu$ for small perturbation of a steady-state water with a big pulse $\varepsilon=0.2$. $T=0.2$ and $N_x=200$. Square: WENOAO-CST; triangle: WENOZQ-CST; circle: WENOMR-CST; plus: WENOJS-XS; solid line: reference solutions.
  • Figure 5: The surface level $h + b$ and discharge $hu$ for small perturbation of a steady-state water with a small pulse $\varepsilon=0.001$. $T=0.2$ and $N_x=200$. Square: WENOAO-CST; triangle: WENOZQ-CST; circle: WENOMR-CST; plus: WENOJS-XS; solid line: reference solutions.
  • ...and 12 more figures

Theorems & Definitions (23)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Example 4.1
  • Example 4.2
  • ...and 13 more