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            <title xml:lang="en">Stationary solutions and well-balanced schemes for the Shallow water model with two velocities</title>
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                <title xml:lang="en">Stationary solutions and well-balanced schemes for the Shallow water model with two velocities</title>
                <title xml:lang="fr">Prise en compte de la topographie dans un modèle de Saint Venant à deux vitesses : solutions stationnaires et schémas numériques</title>
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                <authority type="supervisor">Martin Parisot</authority>
                <authority type="jury">Pascal Omnes [Président]</authority>
                <authority type="jury">Sebastian Noelle [Rapporteur]</authority>
                <authority type="jury">Carine Lucas [Rapporteur]</authority>
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                <term xml:lang="en">Shallow water model with two velocities</term>
                <term xml:lang="en">Steady state solutions</term>
                <term xml:lang="en">Well-balanced schemes</term>
                <term xml:lang="fr">Système de Saint Venant à deux vitesses</term>
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                <term xml:lang="fr">Schémas à l'équilibre</term>
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              <p>In this work, we are interested in the analysis of the "Shallow water model with two velocities". First, we study the steady state solutions using the Bernouilli's principle for C1 regular solutions and the Rankine-Hugoniot relations through discontinuities. Then, we present the types of solutions, their existence and their uniqueness depending on the boundary conditions. Second, we propose several finite volume approximate Riemann solvers for the resolution of the homogeneous Shallow water model with two velocities. The construction of the schemes is based on a recent analysis of the Riemann problem. We present several test cases to illustrate the behavior and the properties of the schemes. Afterwards, we extend these schemes for the model with topography and we propose a suitable numerical approximation of the source term. We prove that the proposed schemes are well-balanced and ensure the positivity of the water heights. Finally, we study the numerical stability of the stationary solutions.</p>
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              <p>L'objectif de ce travail est d'étudier les solutions stationnaires du modèle de Saint Venant à deux vitesses et ensuite de développer des solveurs de Riemann dits "well-balanced" ou équilibre capables de préserver toute solution stationnaire régulière en 1D sous une topographie arbitraire continue. Dans ce but, nous analysons dans un premier temps les solutions stationnaires. Ensuite, nous proposons des solveurs de Riemann pour le système homogène à deux vitesses, i.e sans topographie. Par la suite, nous étendons ces schémas pour prendre en considération le présence de la topographie. Nous validons les différentes parties de notre étude par des résultats numériques.</p>
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