Mathematical Sciences: Computation and Analysis of Waves andTheir Dynamics

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The principal investigator will study select problems in the numerical simulation and analysis of wave propagation. These include both the development of improved methods as well as applications drawn from fluid dynamics, dynamical systems, reaction-diffusion systems and combustion. Problems in methodology to be considered include the construction of accurate and stable boundary conditions at artificial boundaries for computations in unbounded domains, emphasizing error estimation and the equations of fluid dynamics, and bifurcation theory for points in the essential spectrum, with applications to the existence theory of traveling waves in cylindrical domains. The study of these problems makes use of the asymptotic analysis of long range wave propagation, the stability theory of initial-boundary value problems, multiple scales expansions and extensive numerical experimentation. Applications of these methods include the simulation of unsteady compressible flows in exterior and wall-bounded domains, the construction and stability analysis of waves modeling the propagation of curved flame fronts for both thermo-diffusive models and models that include hydrodynamic effects, and a numerical investigation of the existence of neutrally stable helical waves in pipe flow, as predicted by critical layer theory. Wave motion in fluids and other media is a fundamental feature of a variety of disciplines in the sciences and engineering, ranging from meteorology to aeronautics to oil exploration. Despite the diversity of the physics, the science of waves possesses an underlying mathematical unity, and has been the subject of intense and fruitful research. Modern computing technology has provided a new tool for studying these phenomena, allowing a deeper investigation of nonlinear and multidimensional dynamics. The research centers on the development and application of new computational and analytic techniques. It emphasizes the complementary use of computer simulations with other forms of analysis. The particular problems chosen for study arise in fluid dynamics and combustion, but the universal importance of wave propagation makes the work applicable in many other settings.