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An Energy Stable and Well-balanced Scheme for the Ripa System

K. R. Arun, Rahuldev Ghorai

Abstract

We design and analyse an energy stable, structure preserving and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momenta, the pressure gradient and the topography source term. A diligent choice of the interface values of the water height and the temperature ensures the well-balancing property of the scheme for three physically relevant hydrostatic steady states. The explicit in time and finite volume in space scheme preserves the positivity of the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. The results of extensive numerical case studies on benchmark test problems are presented to confirm the theoretical findings.

An Energy Stable and Well-balanced Scheme for the Ripa System

Abstract

We design and analyse an energy stable, structure preserving and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momenta, the pressure gradient and the topography source term. A diligent choice of the interface values of the water height and the temperature ensures the well-balancing property of the scheme for three physically relevant hydrostatic steady states. The explicit in time and finite volume in space scheme preserves the positivity of the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. The results of extensive numerical case studies on benchmark test problems are presented to confirm the theoretical findings.

Paper Structure

This paper contains 23 sections, 8 theorems, 66 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Space Discretisation and Discrete Differential Operators
  3. Mesh and unknowns
  4. Discrete convective fluxes and discrete differential operators
  5. The Finite Volume Scheme
  6. An explicit numerical scheme
  7. Energy stability
  8. Structure preserving property
  9. Well-balancing property
  10. Weak Consistency of the Scheme
  11. Numerical Results
  12. Steady state solution
  13. Lake at rest steady state
  14. Isobaric steady state
  15. Constant height steady state
  16. ...and 8 more sections

Key Result

Lemma 3.2

Any solution to the system eq:dis_updt satisfy the following identity for all $K\in\mathcal{M}$ and $n\in\llbracket0,N-1\rrbracket$: where,

Figures (8)

  • Figure 1: Perturbation in the water height and the velocity.
  • Figure 2: The water height, the velocity, the temperature and the pressure plots for the dam break test case with flat bottom topography at $T=0.2$.
  • Figure 3: The water surface, the velocity, the temperature and pressure plots for the dam break test case with non-flat bottom topography at $T=0.3$.
  • Figure 4: The water surface, the temperature and the pressure plots for the $2d$ steady state problem calculated using the non well-balanced scheme (left) and the well-balanced scheme (right) at $T=0.12$.
  • Figure 5: The water surface, the temperature and the pressure plots for the $2d$ perturbed steady state calculated using the non well-balanced scheme (left) and the well-balanced scheme (right) at $T=0.15$.
  • ...and 3 more figures

Theorems & Definitions (20)

  • Definition 2.1: Discrete mass flux
  • Definition 2.2: Discrete temperature flux
  • Definition 2.3: Discrete momentum flux
  • Definition 2.4: Discrete gradient and discrete divergence
  • Remark 3.1
  • Lemma 3.2: Internal energy identity
  • Lemma 3.3: Kinetic energy identity
  • Lemma 3.4: Potential energy identity
  • Theorem 3.5: Total energy balance of the centred scheme
  • proof
  • ...and 10 more