Numerical Methods for Porous Media and Shallow Water Flows & An Algebra of Singular Semiclassical Pseudodifferential Operators.

Abstract

noindent {bf Numerical Methods for Hyperbolic Systems.} This work considers the Baer-Nunziato model for two-phase flows in porous media with discontinuous porosity. Numerical discretizations may fail to correctly capture the jump conditions across the so-called compaction wave, and yield incorrect solutions. This work formulates the system using the Riemann Invariants across the porosity jump, and proposes a hybrid algorithm that uses the Riemann Invariants formulation across the compaction wave, and the conservative formulation away from the compaction wave. The hybrid scheme is described and numerical results are presented.

The shallow water equations for flows in channels with arbitrary cross-section are also considered. The system forms a hyperbolic set of balance laws. Exact steady-state solutions are available and are controlled by the relation between the bottom topography and the channel geometry. In this work a Roe-type upwind scheme for the system is developed. Considerations of conservation, near steady-state accuracy, velocity regularization and positivity near dry states are discussed. Numerical solutions are presented.

noindent {bf Semiclassical Analysis.} This work is also concerned with the quantization of compact symplectic manifolds with boundary, contained in cotangent bundles. The boundary is always foliated by curves tangent to the kernel of the pull-back of the symplectic form. Assuming the fibrating-and-Bohr-Sommerfeld conditions, an algebra of semiclassical pseudodifferential operators with singular symbols that are naturally associated to the symplectic manifold is constructed. A symbolic calculus is developed, where elements in the algebra have principal symbols on the diagonal and on the flow-out given by the fibers of the foliation, with a singularity on the intersection. The singular principal symbols on the flow-out are identified with families of classical pseudodifferential operators acting on smooth half-densities on the fibers. It is shown that the algebra admits projectors under mild conditions, a generalized Szeg"o limit theorem is proved, and some corresponding singular propagators are studied. The action of the present singular propagators on coherent states is studied numerically, and it is observed that the center of the coherent states may split into more than one when it approaches to the boundary.