Numerical Methods for Balance Laws with Applications to Shallow Water and Multiphase Flows

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This project is concerned with the development, analysis and implementation of numerical methods for balance laws. The focus ison systems which arise by averaging the multidimensional Eulerequations of fluid dynamics. The averaging process producessource terms that are inherently in nonconservation form, whose presence has major consequences for the theory as well as forcomputations of solutions. Computational difficulties include nonuniqueness of solutions, loss of strict hyperbolicity, lack of guiding principles in discretizing nonconservative products and difficulties to compute accurately near steady-state solutions. The aim of this project is to understand, from a numerical view point, how these various characteristics are interconnected, to identify numerical features that need to be respectedand provide design principles which lead to simple and efficientnumerical schemes. This research program will be carried out by studyinga series of nonconservative models, and offer numerical frameworks within which the computational difficulties may be addressed: (i) a relaxation approach addressing the loss of hyperbolicity, and inaccessibility of eigenstructure; (ii) discrete averaging models, giving accurate estimates on closure terms and providing unambiguous approximations for nonconservative products; (iii) hybrid algorithms for two phase flows, which preserves the correct jump conditions across discontinuities in coefficients. As an application, the methods will be used to study two layer exchange flow between reservoirs, and extendedto multilayer shallow water flows through a channel contraction. Analytical predictions are available and will be used for validation.Physical equations of the type considered in this project are used to model a wide range of fluid flows. They are used to model large scale atmospheric and oceanic flows over terrains, and to describe ground flows in porous media. They model river and coastal flows, dam breaks and flooding, such as the flow in the Rigolets strait connecting the Gulf of Mexico and Lake Ponchartrain near New Orleans. They are used in hydraulics, to describe and control exchange flow between reservoirs. In multiphase flows, they model the dynamics of droplet suspensions such as sprays in environmental applications, or cloud dynamics, which may be important in visibility military applications. Alongside experimentsand theory, numerical simulations constitute a major tool of study in these applications areas, and the issues addressed in this project are fundamental to their development. To date, numerical methods for balance laws are still facing major challenges, and provide incomplete and unsatisfactory answers. The directions outlined in this project address these challenges, they are original, they are new and they emphasize general design principles that can be exported to other systems of similar structure. The completion of the proposed project is expected to have a significant impact on computational practices in these communities.