D Lannes, M Rigal

Studies in Applied Mathematics 153 (4), e12751. Download

This paper is devoted to the theoretical and numeri-
cal 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 pro-
cedure that allows one to handle very general linear
or nonlinear boundary conditions. It consists in reduc-
ing the problem to a system of conservation laws with
nonlocal fluxes and coupled to an ordinary differen-
tial equation. 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 bound-
ary conditions would allow one to reconstruct a wave
field from the knowledge of the boundary data only, even
if the initial data are not known.

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