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Bona-Smith-type systems in bounded domains with slip-wall boundary conditions: Theoretical justification and a conservative numerical scheme

Dimitrios Antonopoulos, Dimitrios Mitsotakis

Abstract

Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.

Bona-Smith-type systems in bounded domains with slip-wall boundary conditions: Theoretical justification and a conservative numerical scheme

Abstract

Considered herein is a class of Boussinesq systems of Bona-Smith type that describe water waves in bounded two-dimensional domains with slip-wall boundary conditions and variable bottom topography. Such boundary conditions are necessary in situations involving water waves in channels, ports, and generally in basins with solid boundaries. We prove that, given appropriate initial conditions, the corresponding initial-boundary value problems have unique solutions locally in time, which is a fundamental property of deterministic mathematical modeling. Moreover, we demonstrate that the systems under consideration adhere to three basic conservation laws for water waves: mass, vorticity, and energy conservation. The theoretical analysis of these specific Boussinesq systems leads to a conservative mixed finite element formulation. Using explicit, relaxation Runge-Kutta methods for the discretization in time, we devise a fully discrete scheme for the numerical solution of initial-boundary value problems with slip-wall conditions, preserving mass, vorticity, and energy. Finally, we present a series of challenging numerical experiments to assess the applicability of the new numerical model.
Paper Structure (21 sections, 4 theorems, 106 equations, 15 figures, 8 tables)

This paper contains 21 sections, 4 theorems, 106 equations, 15 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Justification of the mathematical model
  3. Energy conservation and Hamiltonian structure
  4. Existence
  5. Uniqueness
  6. Equivalence between velocity field and potential formulations
  7. Line solitary waves
  8. Dispersion relation
  9. Energy-preserving numerical method
  10. Energy-preserving modified Galerkin method
  11. Time discretization
  12. Computation of initial velocity potential
  13. Generation of solitary waves using the Petviashvili method
  14. Experimental validation
  15. Experimental convergence rates
  16. ...and 6 more sections

Key Result

Proposition 2.1

If $(\eta_0,\phi_0)\in H^2\times H^2$, then there is a time $T>0$ such that the problem (eq:weakform) has a unique weak solution $(\eta,\phi)\in C^1([0,T];H^2\times H^2)$.

Figures (15)

  • Figure 1: Comparison of linear dispersion relations for Bona-Smith systems $2/3\leq \theta^2\leq 1$
  • Figure 2: Reflection of shoaling solitary waves: Initial setup
  • Figure 3: Reflection of shoaling solitary wave of amplitude $A_1=0.07~m$ by a solid wall
  • Figure 4: Reflection of shoaling solitary wave of amplitude $A_2=0.12~m$ by a solid wall
  • Figure 5: Reflection of shoaling solitary waves: Difference $\gamma^n-1$ of the relaxation parameter as a function of $t$
  • ...and 10 more figures

Theorems & Definitions (8)

  • Proposition 2.1
  • proof
  • Corollary 2.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Proposition 3.1
  • proof