Mathematical Sciences: Multilevel and Algebraic Iterative Methods in Large-Scale Computation

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The investigators continue work to develop numerical methods for solving problems of flows in porous media and to develop and analyze fast algorithms for partial differential equations, for computational fluid dynamics, and for several linear algebra problems. Specifically, they (a) develop fast parallel numerical methods for the solution of the Boltzmann equation used to model the transport of neutral and charged particles in 1 and 3 spatial dimensions; (b) develop multilevel zonal techniques for large scale computational fluid dynamics applications on parallel computers; (c) study discretization and solution techniques for fluid flows in porous media that involve complex geometry with heterogeneous material properties; (d) develop a new methodology for the solution of systems of partial differential equations by introducing new dependent variables and forming a least-squares functional. In each case, they exploit and extend multi-level computational techniques. These projects are focused on important applications, such as calculating radiation doses for cancer treatment, ion implantation for semiconductor development, global weather modeling, fluid flow over airfoils and through rocket nozzles, oil recovery and contaminant transport in subsurface water supplies. They reflect the growing recognition that effective numerical modeling of physical phenomena involves several stages - formulation of the mathematical model, transforming the model into a discrete problem, and developing algorithms for the numerical solution of the discrete problem on advanced computers - none of which should be attempted without consideration for the others. Each project examines all components of the process, reformulating the model and discretization when necessary to yield discrete problems that can be efficiently solved numerically on parallel computers.