ABSTRACT
W Wt x , x , ,A( )W 0 0t∈ > (3)
in which the unknown W(x,t) takes values in an open convex set Ω of M, and A is a smooth locally bouded map from Ω to M × M( ). In effect, following (LeFloch 1989), the artificial unknown σ and the equation
σt = 0 (4)
are added to (1), so the augmented system can be written in the form (3) with M = N + 1,
W w
=
⎡ ⎣⎢
⎤ ⎦⎥ ∈ = × ,σ Ω O
and A(W) the matrix-valued function whose block structure is given by
A B( ) ( ) ( ) ( )Df S= −+⎡⎣⎢
⎤ ⎦⎥,0 0
(5)
1 INTRODUCTION
In this work we deal with the numerical approximation of a hyperbolic system of PDE of the form:
w f w S x tt x x x+ , ∈ , > ,w( ) ( )w ( )wB σ 0 (1)
where the unknown w(x,t) takes values in an open convex set of N; f is a regular function from
to N; is a regular matrix function from to N × N( ); S, a function from to N; and σ(x), a
known function from to . System (1) includes as particular cases sys-
tems of conservation laws, when = 0 and S = 0, as well as systems of balance laws, when
= 0. A number of models having this form have been introduced in fluid dynamics to serve as simplified flow models. The system of partial differential equations governing the onedimensional flow of two superposed immiscible layers of shallow water fluids through a straight channel with constant rectangular cross-section (see (Castro et al., 2001)) is a particular case of (1). These systems also appear in models of turbulent shallow water, two-phase flows, sediment transport, turbidity currents, avalanches, submarine avalanches, etc.
