UW Department of Applied Mathematics

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Research Overview

The Applied Mathematics faculty are actively involved in many areas of research, including the following:  

Asymptotic and Perturbation Analysis

R. E. O'Malley, Jr.

Perturbation methods are used broadly in science to obtain approximate solutions based on known exact solutions to nearby problems. Such asymptotic techniques can also be used to provide initial guesses for numerical approximations, and they can now be generated through clever use of symbolic computation. Members of the department have most effectively used singular perturbation methods to analyze problems in fluid and solid mechanics, control theory and celestial mechanics, and a variety of nonlinear oscillation, nonlinear wave propagation, and reaction-diffusion systems arising in numerous physical and biological contexts. Their mathematical study seeks to develop generally applicable methods, based on our expanding understanding of matching, multiscale, and regularization concepts. A variety of challenging applied problems continue to motivate such analysis.

 

Classical Analysis

W. O. Criminale, R. E. O'Malley, Jr.

Underlying many areas of Applied Mathematics is a framework of analysis, erected over many years. Elements of this framework include real and complex function theory, ordinary and partial differential equations, integral equations, differential geometry, tensor analysis, special functions, variational methods, and probability theory. Department members continually use, and sometimes further develop, all of these tools in their research.

 

Dynamical Systems

M. Kot, B. Deconinck, J. Nathan Kutz

Dynamical systems theory offers an alternative point of view to aspects of the theory of differential equations. In this theory the focal point is typically the study of individual solutions. In dynamical systems, one studies the collection of all solutions as a whole, leading to a more global viewpoint. This is the natural framework for bifurcation analysis, and the study of chaos. The systems may be conservative or dissipative, continues or discrete, finite or infinite dimensional. Dynamical systems arise in virtually any application area. Different members of the department use methods from dynamical systems in a variety of applications, including fluid mechanics, mathematical biology and nonlinear optics.

 

Geophysical Fluid Dynamics

C. S. Bretherton, W. O. Criminale, P. J. Schmid, K. K. Tung

Mathematical models play a crucial role in our understanding of the fluid dynamics of the atmosphere and oceans. Our interests include mathematical methods for studying the hydrodynamical instability of shear flows, transition from laminar flow to turbulence, applications of fractals to turbulence, two-dimensional and quasi-geostrophic turbulence theory and computation, and large-scale nonlinear wave mechanics. We also develop and apply realistic coupled radiative-chemical-dynamical models for studying stratospheric chemistry, and coupled radiative-microphysical-dynamical models for studying the interaction of atmospheric turbulence and cloud systems These two topics are salient for understanding how man is changing the earth's climate. Our work involves a strong interaction of computer modelling and classical applied analysis. This research group actively collaborates with scientists in the Atmospheric Science, Oceanography, and Geophysics department, and trains students in the emerging interdisciplinary area of earth system modeling, in addition to providing a traditional education in classical fluid dynamics.

 

Mathematical Biology

Mark Kot, H. Qian

Mathematical biology is an increasingly large and well-established branch of applied mathematics. This growth reflects both the increasing importance of the biological and biomedical sciences and an appreciation for the mathematical subtleties and challenges that arise in the modelling of complex biological systems. Our interest, as a group, lies in understanding the spatial and temporal patterns that arise in dynamic biological systems. Our mathematical activities range from reaction-diffusion equations, to nonlinear and chaotic dynamics, to optimization. We employ a variety of tools and models to study problems that arise in development, epidemiology, ecology, resource management, and biomechanics; and we maintain active collaborations with a large number and variety of biologists and biomedical departments both in the University and elsewhere.

 

Nonlinear Waves

R. J. LeVeque, B. Deconinck, J. N. Kutz

Most problems in applied mathematics are inherently nonlinear. The effects due to nonlinearities may become important under the right circumstances. The area of nonlinear waves considers how nonlinear effects influence problems involving wave propagation. Sometimes these effects are desirable and lead to new applications (nonlinear optics), other times one has no choice but to consider their impact (water waves). The area of nonlinear waves encompasses a large collection of phenomena, such as the formation and propagation of shocks and solitary waves. The area received renewed interest starting in the 1960's with the development of soliton theory, which examines completely integrable systems and classes of their special solutions.  

Numerical Analysis and Scientific Computation

L. M. Adams, C. S. Bretherton, R. J. LeVeque, P. J. Schmid, J. Nathan Kutz

Many practical problems in science and engineering cannot be solved completely by analytical means. Research in the area of numerical analysis and scientific computation is concerned with the development and analysis of numerical algorithms, the implementation of these algorithms on modern computer architectures, and the use of numerical methods in conjunction with mathematical modeling to solve large-scale practical problems. Major research areas in this department include computational fluid dynamics (CFD), interface and front tracking methods, iterative methods in numerical linear algebra, and algorithms for parallel computers.

Formal training in numerical analysis is coordinated with the Mathematics Department and students from both departments take the same set of graduate courses in this area and jointly run the Numerical Analysis Journal Club. Students in Applied Mathematics are also expected to become proficient in an outside application area, and often work on specific applied problems.

Current research projects in CFD include the development of high resolution methods for solving nonlinear conservation laws with shock wave solutions; numerical methods for atmospheric flows, particularly cloud formation; Cartesian grid methods for solving multidimensional problems in complicated geometries on uniform grids; and the study of fluid stability problems with spectral methods. Front tracking methods for fluid flow problems with free surfaces or immersed interfaces are being studied in the context of porous media flow (ground water or oil reservoir simulation) and also in physiological flows with elastic membranes. Nonequilibrium flows in combustion and astrophysical simulation are also of interest. Immersed interface methods are also being developed for solidification or melting problems and for seismic wave equations with discontinuous coefficients that arise in modeling the geological structure of the earth.

Another research focus is the development of methods for large-scale scientific computations that are suited to implementation on parallel computer architectures. Current interests include preconditioners for the iterative solution of large linear or nonlinear systems, methods for the symmetric and nonsymmetric eigenvalue problems, and methods for general interface problems in complicated domains. The actual implementation and testing of methods on parallel architectures is possible through collaboration with the Department of Computer Science, the Boeing Company, and the Pacific Northwest Labs.

Faculty members in this group have strong ties with researchers in specific applications areas both within this department and in other departments on campus, including Aeronautics and Astronautics, Astronomy, Atmospheric Science, Chemical Engineering, Civil Engineering, Computer Science, Geology Mechanical Engineering, and Zoology. They are also active in the international community and have ongoing collaboration with researchers at other universities and laboratories throughout the world.

 

Stability and Control of Fluid Flow

P.J. Schmid, W.O. Criminale

The stability of fluid flow is one of the central fields of fluid mechanics. Many flows exhibit instabilities of various forms as a governing parameter is varied: mixing layers roll up into large eddies, a street of vortices forms behind bluff between two plates, vortical structures appear near concave walls, and characteristic corrugations develop on smoke rings. These instabilities are often the precursor of a transition process into more complex fluid motion and ultimately into turbulence. Understanding the rise of instabilities and the mechanisms underlying the transition from laminar to turbulent flows has occupied fluid dynamicists over many decades. Hydrodynamic stability theory has matured from early attempts using perturbation theory and asymptotics to a modern discipline firmly based on spectral theory and numerical computations. Especially the advent of computers has fueled the investigation of increasingly complex - and realistic - flow configurations. Over the last decade, the analysis of fluid instabilities from a nonmodal point of view has made significant progress, and many transition processes are now better understood using this approach.

Long before the means and technology were available there has been the desire to manipulate the natural behavior of fluid flow. Whether it is the suppression of instabilities to extend the prevalence of laminar flow, whether it is the reduction of skin friction and drag to accomplish a more efficient operation of high-speed vehicles, or whether it is the amplification of existing instability mechanism to enhance mixing or heat transport, the response of fluid behavior to external forcing has to be understood in order to design efficient and effective control strategies that achieve the desired goal. The field of flow control lies at the interface of hydrodynamic stability theory and control theory: optimal control theory, robust control theory, controllability, estimation and model reduction are now concepts readily applied to fluid flow systems. Continuous (based on adjoint equations) and discrete (based on Riccati equations) formulations are equally used in flow control, and the results show great promise and technological reward. Like in stability theory, the use of direct numerical simulations has brought a valuable component to the field that allows the design of control strategies for three-dimensional flows in realistic geometries.

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