Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
Chapitre 3
Finite Volume discretization and
numerical simulations of the
multilayer shallow water model
In this chapter, we propose a discretization of system (2.1.10) with a Finite Volume
method in Section 3.1. Then we present some numerical results in Section 3.2: we compare
the model to classical shallow water models as well as with the primitive system when there
is no viscosity. Moreover, we illustrate the dynamic behavior of our model, with additions
and substractions of layers.
3.1 Numerical scheme
We present in this section the discrete version of the system (2.1.10). Several strategies
are possible. For example, in [12, 13, 58], the authors perform an upwind scheme based on
approximate Riemann state solvers. In [14, 15], the authors use a kinetic formulation. Here,
we will simply use a Finite Volume scheme [133, 134], by isolating an hyperbolic part of
the system, for which we can evaluate exact eigenvalues, without computing eigenvectors.
The lawfulness of this choice can be discussed but the results obtained are good, and the
simplicity of the scheme makes it easy to implement, while the code is dynamic: we can
add or remove layers when the uppest layer height becomes too large or too small.
In order to design a numerical scheme, we will consider a third formulation of the one
dimensional multilayer problem (2.1.10). This time, the unknowns are denoted by V, lying
in R N+1 , and W in R N :
=
Vi 0iN = (h,huN,u1,...,uN−1) T , W = (w1/2,...,w N−1/2) T .
Hence, we separate the viscous terms: the horizontal one is included in the flux term
with respect to x, and the vertical one is kept in the source term. The formulation reads
as:
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