General boundary conditions for a Boussinesq model with varying bathymetry

TL;DR

This work tackles the 1D Boussinesq-Abbott equations with non-flat bottom by introducing a nonlocal-flux reformulation that reduces the IBVP to a simple evolution problem coupled to an ODE for boundary traces. It develops two well-balanced, hybrid finite-volume/finite-difference schemes (first- and second-order) capable of handling very general time-dependent boundary conditions expressed via input/output boundary functions. Theoretical results establish well-posedness for flat and topographic bottoms, while numerical experiments demonstrate accurate wave resolution, boundary-condition flexibility, and partial evidence of asymptotic stability when boundary inputs align with shallow-water Riemann invariants. The work also outlines a path to coupling with higher-fidelity models and discusses practical implications for coastal-oceanographic simulations and boundary-data-based wave-field reconstruction.

Abstract

This paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq-Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ODE. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data is not known.

Figures (5)

• Figure 1: Free surface elevation for the incoming and outgoing solitary wave test-cases (respectively top and bottom); incoming Riemann invariants imposed at the boundaries
• Figure 2: Left: initial condition. Right: reference solution, Lax-Friedrichs and MacCormack approximations at time $t = 15$, both obtained using the same CFL constant of 0.45.
• Figure 3: Initial elevation for reference and small domain solutions ($\mu=1$).
• Figure 4: Flow over bar test-case for $\mu=1$ (first column) and $\mu=10^{-1}$ (second column). First and second rows correspond respectively to the MacCormack elevation and its deviation from the reference elevation $\widetilde{\zeta}$ at final time $t = 50\, T_0$ for various boundary conditions. The last row represents the $\ell^2$ error \ref{['eq:L2-error']} over time. Out of the three boundary conditions investigated here, only enforcing the incoming Riemann invariants allows this error to decay towards zero.
• Figure 5: Flow over bar test-case for $\mu=10^{-2}$. The first plot was obtained at final time $t = 50\, T_0$. Enforcing $\zeta$ or $q$ as boundary conditions led to a blow up of the MacCormack approximation, therefore only the case of incoming Riemann invariants is displayed here. The $\ell^2$ error \ref{['eq:L2-error']} decays over time but does not quite vanish; this is most likely due to the coarseness of the mesh which make it difficult to accurately capture the high frequencies that develop in the solution. A fix would be to refine the mesh further.

Theorems & Definitions (32)

• Remark 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• Remark 2.6
• proof
• Lemma 2.7