Chi-Wang Shu - US grants

Shu 9214488 The objective of this proposed research is to develop accurate and stable algorithms, based on the essentially non-oscillatory (ENO) methodology, which were originally designed for general hyperbolic conservation laws and in particular for the compressible Euler equations in gas dynamics, to device simulations in two and eventually in three space dimensions. The emphasis will be on simulating the hydrodynamioc (HD) model and the recently developed Energy-Transport (ET) model and investigating their physical applications. Issues related to device simulations, such as the choice of boundary conditions based upon physical as well as numerical considerations, the adequate handling of boundary and inner layers (high gradient regions), the adequate handling of mild singularities in the potential due to the boundary conditions, the adequate handling of possibly stiff source terms, and parallel implementation of the algorithm will be investigated. The eventual goal of the effort is to develop a numerical algorithm which is accurate and robust, suitable for device simulations using not only the HD and ET models but also other models as well. Physical applications of related models will also be investigated in collaboration with other applied mathematicians, device physicists and engineers. ***

9809815 Karniadakis A new class of projection methods, the Discontinuous Galerkin Methods (DGM), has been developed recently and has found its use very quickly in such diverse applications as aeroacoustics, semi-conductor device simulation, turbomachinery, turbulent flows, materials processing, MHD and plasma simulations, and image processing. While there has been a lot of interest from mathematicians, physicists and engineers in DGM, only scattered information is available and there has been no prior effort in organizing and publishing the existing volume of knowledge on this subject. The authors of this proposal plan to organize the first international symposium on DGM with equal emphasis on the theory, numerical implementation, and applications. They plan to publish a book to serve as the first reference on this subject with both review articles and state-of-the-art contributions presented at the symposium. The Symposium will include both invited papers as well as other contributions in order to encourage wider participation especially from young scientists and under-represented minorities.

The objective of the proposed research is to study high order numerical
methods for shock calculations and for computational electromagnetics.
High order methods can resolve complicated physical problems with a
relatively coarse mesh, hence reducing computational cost for such
problems. However, high order methods for shock calculations and for
computational electromagnetics involve many theoretical and practical
issues which must be investigated. The proposed work involves the
development, analysis, and applications of high order finite difference,
finite element and spectral methods in the applications areas of
computational fluid dynamics and computational electromagnetics.
Two emphasized aspects of the proposed effort are complex geometries
and efficient parallel implementations. In particular, the proposal contains the following components: high order methods for shock wave calculations, including finite difference ENO and WENO schemes, finite element discontinuous Galerkin methods, spectral methods, and high order methods in computational
electromagnetics, including spectral methods, spectral multidomain
methods, and absorbing layers. It is expected that the proposed effort will improve the state of art in high order methods for discontinuous problems and long time integrations, especially in complex geometry.

9815423
Strauss

This three-year award for US-France cooperative research in applied mathematics involves Walter A. Strauss, C. Shu and Yan Guo of Brown University and Pierre Degond of the Universite Paul Sabatier and Frederic Poupaud of the Universite de Nice. The project is aimed at the design, development, analysis and numerical simulation of hybrid quantum kinetic models and their application to the simulation of quantum semiconductor devices. In many applications, the quantum effects are important only in subregions of the device, while the phenomena in a larger domain are governed by classical transport. The objective of this project is to design models that are able to switch from a classical description to a quantum description. The US investigators bring to this collaboration expertise in classical modeling and simulation. This is complemented by French expertise in quantum models.

This proposal is a continuation of the research supported by NSF Grant ECS-9627849. The objective is to study the modeling, mathematical analysis, and in particular novel, reliable and efficient numerical methods for semiconductor device simulations. The emphasis of the proposed effort will be on the analysis and simulation of the recently developed high field (HF) models without and with energy transport, the hydroynamic (HD) models, the kinetic models, the energy transport, (ET) models, the quantum related models, and optoelectronic device models involving the coupling with the Maxwell equations. Both time accurate models and numerical methods and steady state solutions will be considered. Collaborations and communications with device physicists, engineers, and applied mathematicians, will be maintained, to investigate device simulation problems from physical, mathematical, and numerical points of view.

The Division of Applied Mathematics at Brown University will purchase a high-end multi processor computing and visualization facility suitable for computationally intensive computing, postprocessing, and visualization of large data sets, which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects, including in particular: Numerical Methods for Deterministic Optimal Control Problems, NSF Grant DMS-9704426 to Dupuis; High Order Methods for Shock Calculations and Computational Electromagnetics, NSF Grant DMS-9804985 to Shu and Gottlieb; Modeling Natural Image Statistics, NSF Grant DMS-9615444 to Mumford; Invariant Manifolds and Complex Behavior in Nonlinear Physical Systems, NSF Grant DMS-9800922 to Haller; and Continuum Physics and Systems of Conservation Laws, NSF Grant DMS-980352 to Dafermos.

Abstract
Brown

The Mathematics Department and the Division of Applied Mathematics at Brown University will undertake a program of integration of research and education, supported by a grant from the NSF VIGRE Program. We will develop activities on the undergraduate, graduate and postdoctoral level which will strengthen and provide better coordination for the mathematical sciences program of the two departments. Through the participation and support of the members of the Mathematics and Applied Mathematics departments, in particular VIGRE Postdoctoral Instructors, VIGRE Graduate Trainees and Brown undergraduates, our program will forge scientific communication and research interaction between participants at all levels of mathematical experience and seniority.
The VIGRE program is an opportunity for the Mathematics Department and the Division of Applied Mathematics to review and reshape their graduate and undergraduate programs, responding to changes in the discipline, research priorities and the evolving landscape of career opportunities for Mathematics degree holders with experience in research. We plan to rethink our programs, broaden their scope and deepen their contributions to graduate education, without enlarging the size of the mathematical sciences program at Brown. Our main objectives are to develop innovative activities in our programs which will serve (1) to provide modes of scientific interaction between practicing research mathematicians and students of mathematics at all levels, (2) to give our graduate and undergraduate students more in-depth contact with areas and material of current research interest, and (3) to produce graduate students who are more scientifically literate and more articulate, and who possess the basic skills for a wider range of career possibilities after completing their degree.

This program will support 12 graduate students as VIGRE graduate trainees (GTs), 3 VIGRE postdoctoral fellows (VPs) and up to 12 undergraduate research projects (UTRAs) at a time. These resources will be shared equally between the two mathematics departments as described below. The VIGRE program at Brown will consist of the following components:
(1) a graduate traineeship program (GTs)
(2) an undergraduate tutoring and research program (UTRAs)
(3) a postdoctoral fellowship program (VPs)
(4) support for a mentoring program for women in sciences (WiSE) and under-represented minorities in sciences (ExSEL)
(5) a graduate and peer tutoring resource center in mathematics for undergraduates (MRC)
(6) a program of outreach to gifted high school students at local area high schools
(7) a program of outreach for disadvantaged high school students, resumed education students and their teachers
(8) a program of internships with industry and with the National
Laboratories

Graduate Traineeships (GTs) sponsored by VIGRE are opportunities for students to complete a graduate education in mathematics in a stimulating and research productive environment, with a balanced number of research and teaching semesters, and with adequate summer support. Postdoctoral Fellows (VPs) will have the opportunity to work closely in research groups, with reduced teaching load and with close contact with both senior faculty and graduate students.
Principal features of the graduate, undergraduate and postdoctoral programs include:
(1) A series of VIGRE graduate seminars, modeled on the example of a 'literature seminar' and/or a 'lab meeting' that is common in experimental sciences.
(2) Additional advanced graduate/research level seminars, on a broad set of current topics in mathematics and applied mathematics.
(3) A seminar course series in the graduate program focusing on scientific communication and effective mathematics instruction.
(4) A focus on individual undergraduate research activities (UTRAs), directed by faculty and VIGRE Postdoctoral fellows.
Resources for these activities will come both from the VIGRE program grant and from an increased level of coordination and cooperation between the Mathematics Department and the Division of Applied Mathematics.
Graduate Trainees supported by VIGRE will expect support for five years of study, and teach between two and four semesters during their tenure at Brown. They will also participate in the VIGRE activities outside of their usual course work, which they will be directly involved in coordinating. Activities involving undergraduates include (1) UTRA research seminars, (2) mentoring activities with the WiSE women in sciences undergraduate support group, and with the ExSEL minority sciences support group, and (3) organization and participation in the MRC, mathematics drop-in resource centers for undergraduates. Activities which involve the local community include (4) participation in outreach programs to Providence area schools, aimed both for gifted and for disadvantaged students and for teachers, and programs in community continuing education centers.
Graduate Trainees will be encouraged to broaden their scientific background, past the confines of their particular research specialization and across department boundaries. In particular the VIGRE program intends to make the interface at Brown between Mathematics and Applied Mathematics substantially more permeable. Furthermore, there will be regular programs of summer internships with industry and national labs.
VIGRE Postdoctoral Fellows will benefit from the strong research environment in both the programs in mathematics and applied mathematics, and they will be expected to contribute to the VIGRE graduate seminars. They will be required to teach one course each semester that they are on campus. They will also have opportunities to collaborate with research staff at national labs and in industry, and they are expected to benefit from communication with the Brown departments of Chemistry, Computer Science, Engineering, Physics, and the campus-based group in cognition (computer and natural vision, artificial and natural intelligence).
Objectives: We intend to provide our finishing PhDs with a broader view of mathematics and science, with perspective on the relationship of their particular discipline within it. They should be aware and prepared for the many possible careers both in and outside of academia that are open to degree holders with mathematics research experience. Graduate students will receive training in scientific communication and in mathematics instruction. We intend to increase the quality of our incoming graduate classes, through a focussed identification and recruitment program, and to increase the proportion of women and under represented minority students in our program. The Brown VIGRE program will pay particular attention to the problem of retention of these students. VIGRE resources will increase the number of research support semesters for participants, and as a result we expect to lower the average time to degree. We intend to produce better prepared undergraduate mathematics concentrators, who already have some research experience, many of whom are expected to go on to graduate school and subsequently a career in the mathematical sciences. We expect our VIGRE Postdoctoral Fellows to have an additional competitive advantage in the academic job market, from the breadth of their research experience at Brown, from the increased opportunity for collaborative research, from the availability of contacts with industry and national labs, and from the focus on instruction and communication of mathematics, and their experience teaching at a high quality undergraduate institution.
Impact: We expect that we will be able to increase the numbers of American undergraduate majors in mathematics, and to increase the subset of the pool which goes on to graduate school in mathematics. We expect to increase the proportion of Am

Shu
0207451
The investigator and his colleague study high-order-accuracy
computational methods for linear and nonlinear waves, with
emphasis on shock wave calculations and simulation of
electro-magnetics waves. The work includes the development and
analysis of high-order finite difference, finite element, and
spectral methods, as well as applications of these methods to
computational fluid dynamics and computational electro-magnetics.
In particular, the efforts include the following components:
high-order methods for shock wave calculations, including finite
difference WENO schemes, spectral methods for supersonic reactive
flows, and finite element discontinuous Galerkin methods;
high-order methods for Maxwell's equations; and perfectly matched
absorbing layers.
Computers are now used more extensively in engineering and
other applied sciences. An important component to effectively
use computers is the design and analysis of efficient numerical
algorithms. Important applications such as computer-aided design
of aircraft are to a large extent dependent on efficient and
reliable algorithms designed by applied mathematicians. The
high-order methods the investigators develop and analyze in this
project should significantly increase the efficiency and
reliability of algorithms used in crucial application areas such
as aerospace industry, communications, and material science.

ABSTRACT

FRG: Multi-Dimensional Problems for the Euler Equations
of Compressible Fluid Flow and Related Problems
in Hyperbolic Conservation Laws

Historically, fluid and solid mechanics study the motion of
incompressible and compressible materials, with or without internal
dissipation. For gases and solids with internal dissipation as a
secondary effect, the gross wave dynamics is governed by inviscid,
thermal diffusionless, dynamics. Within these categories, compressible
motion for solids corresponds to the study of elastic waves and their
propagation; compressible motion for fluids is usually associated with
inviscid gas dynamics. Furthermore both compressible solids and
fluids exhibit shock waves and hence we must search for discontinous
solutions to the underlying equations of motion.
Incompressible motion on the other hand concerns
itself with the motion of denser fluids where the idealization of
incompressibility is useful, e.g. water or oil, as well as the motion of
certain solids like rubber. While there are still many important
mathematical issues to be resolved for incompressible fluids, for example,
the well-posedness of the Navier-Stokes equations in three space
dimensions, the mathematical study of compressible
solids (as represented by the equations of nonlinear elastodynamics) and
fluids (as represented by the Euler equations of inviscid flows)
in two and three space dimensions is even less developed.
This provides the motivation to the proposers to collaborate in a
three year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics
and related problems in elastodynamics.
The core of our plan is to arrange a sustained interaction between and
around the members of the group, who will
(1) collaborate scientifically, focusing on the advancement of the
analysis of multi-dimensional compressible flows by developing new
theoretical techniques and by using and designing effective, robust and
reliable numerical methods;
(2) work together over the next several years to create the environment
and manpower necessary for the research on multi-dimensional compressible
Euler equations and related problems to flourish; and in the meantime,
(3) share the responsibility of training graduate students and
postdoctoral fellows.

The project is devoted to a mathematical study of the Euler equations
governing the motion of an inviscid compressible fluid and related
problems. Compressible fluids occur all around us in nature, e.g. gases
and plasmas, whose study is crucial to understanding aerodyanmics,
atmospheric sciences, thermodynamics, etc.
While the one-dimensional fluid flows are rather well understood, the
general theory for multi-dimensional flows is comparatively mathematically
underdeveloped. The proposers will collaborate in a three
year effort to advance the mathematical understanding of the
multi-dimensional equations of inviscid compressible fluid dynamics.
Success in this project will advance knowledge of this fundamental area of
mathematics and mechanics and will introduce a new generation of
researchers to the outstanding problems in the field.

The investigators propose to organize the next International Conference on "Spectral And High-Order Methods (ICOSAHOM'04)" on June 21-25 2004 at Brown University. The algorithmic themes addressed in this conference are many and new, and attract most leading researchers in spectral methods, h-p finite elements, ENO schemes, high-order finite differences, and wavelets. There are nine confirmed keynote speakers (selected by voting by the scientific committee), four of whom are from USA, and two are women. The topics they will cover are: wavelets, h-p finite elements, singular solutions, electromagnetics, non-Newtonian flows, inverse problems, uncertainty, ocean modeling, fast solvers, and high-order finite differences.

Spectral and high-order methods have become increasingly important in diverse and important applications as aeroacoustics, electromagnetics and optics, ocean and climate modeling. These methods continue to grow in importance as a simulation tool and attract global attention. The proposed conference, part of a long series, continues to be the main meeting where such methods are discussed and new and exciting applications displayed. Furthermore, emphasis is traditionally placed on involving many young researchers as well as underrepresented minorities and women. As a new thing, the conference will conclude with a round-table discussion consisting of a mix of invited speakers, minisymposia organizers, and researchers from national labs and industry. The objective is to discuss and report to NSF the open issues and future algorithmic and application trends of high-order and other discretization methods and their potential in applications of national interest.

ABSTRACT

The Division of Applied Mathematics at Brown University is acquiring research computing equipment dedicated to support of research in the mathematical sciences. The main items to be purchased include: (1) high-bandwidth low-cost storage optimized for use with the Lustre file system and existing Linux clusters, (2) high-speed network components to support trunked-ethernet communications between the compute nodes, storage, and graphics systems, (3) high-end Linux client PCs for data analysis and visualization, and (4) a small computer experimental facility to support collection and analysis of neural spike-train data for the mathematical modeling and statistical analysis of biological vision. The computing equipment will be used to carry out the research of NSF-sponsored projects in numerical methods for partial differential equations, mathematical modeling in brain science, and statistical procedures in experimental neuroscience.

The research in applied mathematics is closely tied to applications in engineering, physics, and the life sciences. The NSF-sponsored research focuses on development of new mathematical and statistical methods designed for the application areas. For example, some of the work on computational methods for engineering is motivated by the need to design stealth aircraft. Also, the research on new statistical methods is needed for the advancement of brain science, to understand how information is encoded in neural spike trains. The computer equipment provided by this grant is crucial for the ability of faculty and students to create these new mathematical and statistical methodologies.

Many physical, biological, and engineering problems involve linear or nonlinear wave phenomena in adaptive, multiscale, and uncertain environments. The focus of this proposal is to design, analyze and apply high-order accurate and highly efficient numerical algorithms, including high-order weighted essentially non-oscillatory methods, high-order discontinuous Galerkin finite element methods, and spectral methods, for effective simulations of such wave phenomena using computers. Mathematical tools will be used to guide the design of such algorithms so that they will be able to produce reliable and accurate results for such complicated wave phenomena with a high speed and hence a fast turnaround time. This will in turn allow a deeper understanding of the physical and biological phenomena and also to help in many engineering designs, such as the design of aircrafts and semiconductor devices. In this project, problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues for the computation on massively parallel computers will be addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists will enable the efficient application of these new algorithms or new features in existing algorithms. The training of young researchers in this area will also be an important component of this project.

AST-0506734
Shu

This project will attempt to combine algorithm development for high order accurate, weighted essentially non-oscillatory schemes for solving high speed, high Mach number inviscid flows containing strong shocks and other complex flow structures, with cosmological applications in hydrodynamic simulations. The objective is to develop efficient and robust numerical methods for cosmological applications, and to consider the hydrodynamics of the epoch of reionization, the hydrodynamical behavior of baryon gas around dark matter singularities, the statistical discrepancy between the intergalactic medium and dark matter in the non-linear regime of clustering, the entropy production of gravitational shocks, and other complex structural problems. Addressing cosmology's unique difficulties, such as the existence of extremely strong singularities and the effective coupling of hyperbolic hydrodynamic equations with elliptic equations for self-gravity, requires advances in computational mathematics and scientific computing.

Broader impacts include enhancing a collaboration across two universities, advancing the mathematical sophistication of astrophysics, and making algorithms and codes of general utility available to the wider astrophysics, physics, and engineering communities.

In this project, research in the algorithm design and analysis
of high order numerical methods, including the finite difference
and finite volume weighted essentially non-oscillatory (WENO)
schemes, discontinuous Galerkin finite element methods, and
particle methods, for hyperbolic and other convection dominated
partial differential equations, especially in adaptive, multiscale
and uncertain environments, will be carried out. Parallel
implementation and applications of these methods will also be
addressed. The intellectual merit of the proposed activity lies
in its comprehensive coverage of algorithm development, analysis,
implementation and applications. Problems in applications
motivate the design of new algorithms or new features in
existing algorithms; mathematics tools are used to analyze
these algorithms to give guidelines for their applicability
and limitations; practical considerations including parallel
implementation issues are addressed to make the algorithms
competitive in large scale calculations; and collaborations
with engineers and other applied scientists enable the
efficient application of these new algorithms or new features
in existing algorithms.

The proposed research aims at the design of efficient algorithms,
which, when used on today's powerful computers, will help to
solve many problems from diversified applications such as
aerodynamics and aeroacoustics for aircraft design,
electromagnetism wave simulation for communications,
and semiconductor device simulation for the computer industry.
The thrust of this proposal is to use powerful mathematical
tools to guide the design of algorithms, so that they are more
efficient, more reliable, and more robust in applications.

This purpose of this grant is to support an International Conference
on Advances in Scientific Computing at Brown University on December
6-8, 2009, to honor the memory of Professor David Gottlieb and to
review recent advances and explore exciting new directions in
scientific computing and related numerical solution of partial
differential equations and mathematical modeling for time dependent
problems and their applications. A notable feature of this
conference will be the emphasis on the crucial role of significant
mathematics in the design of advanced algorithms applicable to real
world problems. The conference will include invited speakers
ranging from numerical analysts with a strong interest in
applications, to applied and computational mathematicians to
engineers, physicists and scientists in other fields. The list of
invited speakers includes both very senior leaders in the field
and relatively young scientists. Represented in this list are both
women and minority mathematicians.

The proposed conference will be an effective venue for the exchange
of ideas among a diversified list of invited speakers and
participants. It will provide valuable guidance to young
participants to identify promising problems and application areas.
It is expected that this conference will push forward the research
in algorithm design and applications utilizing significant
mathematics to improve efficiency and effectiveness. These
algorithms have been and will continue to be widely used in
applications which are of national interest, including homeland
security (image processing and pattern recognizing), environment,
aerospace engineering, communications, energy science, and climate
modeling.

This award provides funds to secure computer equipment with initial setup and maintenance to support the computational requirements of several NSF supported projects, including algorithm design and application of high order numerical methods for convection dominated problems and high order numerical methods for elasticity, and numerical relativity. The algorithms being investigated include the finite difference and finite volume WENO schemes, discontinuous Galerkin and other finite element methods, and spectral methods, especially in adaptive, multiscale and uncertain environments. A particular emphasis of some of the proposed work is development and algorithmic adaptation of high-order methods to graphics processing unit (GPU) accelerated architectures.

The algorithms developed in this project will help to improve our capability to understand the fundamental physics of many important problems, and to facilitate the engineering design and production of advanced materials, aircrafts, semiconductor devices, and energy products.

In this project, research in the algorithm design and analysis of high order numerical methods, including the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations, will be carried out. While the emphasis of this project is on algorithm design and analysis, close attention will be paid to efficient parallel implementation and applications. The intellectual merit of the proposed activity lies in its comprehensive coverage of algorithm development, analysis, implementation and applications. Problems in applications motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new featuresin existing algorithms.

The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications with involving convection dominated partial differential equations, in adaptive, multiscale and uncertain environments. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications. The application areas include (but are not limited to) computational fluid dynamics, traffic flow problems, semiconductor device simulations, and computational biology. Graduate students will be involved in this project, and will get training in performing mathematics research on problems closely related to applications. Special attention will be paid to the recruitment and training of Ph.D. students from under-represented groups including women.

In this project, the PI will perform research in the algorithm design and analysis of high order numerical methods. These algorithms are used to solve scientific and engineering problems arising from diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biological problems. Even with today's fast computers, it is still essential to design efficient and reliable algorithms which can be used to obtain accurate solutions to these application problems. The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications mentioned above. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications.

The algorithms to be investigated include the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations (PDEs). While the emphasis of this project is on algorithm design and analysis, close attention will be paid to applications. Topics of proposed investigations will include the study on high order accurate bound-preserving algorithms and applications, an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on rectangular meshes when the physical boundary is not aligned with the meshes, WENO schemes with subcell resolution for nonconservative problems, Lagrangian type finite volume schemes for multi-material flows, energy-conserving discontinuous Galerkin methods for long time simulation of wave problems, efficient discontinuous Galerkin methods for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications in adaptive computation, simple WENO limiters for discontinuous Galerkin methods in unstructured meshes for problems with strong shocks, multi-scale methods based on the discontinuous Galerkin framework, analysis and numerical solutions for traffic and pedestrian flow models, turbulence simulation in cosmology, and study on aggregation and coordinated movement in computational biology. Problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new features in existing algorithms.

In this project the PI will perform research in algorithm design and analysis of high order accurate and efficient numerical methods for solving partial differential equations. These algorithms are used to solve scientific and engineering problems arising from diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biological problems. Even with today's fast computers, it is still essential to design efficient and reliable algorithms which can be used to obtain accurate solutions to these application problems. The broader impacts resulting from the proposed activity will be a suite of powerful computational tools, suitable for various applications mentioned above. These tools are expected to make positive contributions to computer simulations of the complicated solution structure in these applications.

The algorithms the PI plans to investigate include the finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin finite element methods, for solving hyperbolic and other convection dominated partial differential equations (PDEs). While the emphasis of this project is on algorithm design and analysis, close attention will be paid to applications. Topics of proposed investigations will include the study on an inverse Lax-Wendroff procedure for high order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, Lagrangian type finite volume schemes for multi-material flows, a simple weighted essentially non-oscillatory limiter for discontinuous Galerkin methods with strong shocks, high order stable conservative methods on arbitrary point clouds, discontinuous Galerkin methods for weakly coupled hyperbolic multi-domain and network problems, efficient time-stepping techniques for discontinuous Galerkin schemes, high order accurate bound-preserving schemes and applications, bound-preserving high order discontinuous Galerkin schemes for radiative transfer equations, energy-conserving DG methods for Maxwell's equations in Drude metamaterials, efficient discontinuous Galerkin method for front propagation problems with obstacles, superconvergence analysis of discontinuous Galerkin methods and its applications, multi-scale methods based on the discontinuous Galerkin framework, and applications in areas including traffic and pedestrian flow models and aggregation, coordinated movement and cell proliferation in computational biology. Problems in applications will motivate the design of new algorithms or new features in existing algorithms; mathematics tools are used to analyze these algorithms to give guidelines for their applicability and limitations; practical considerations including parallel implementation issues are addressed to make the algorithms competitive in large scale calculations; and collaborations with engineers and other applied scientists enable the efficient application of these new algorithms or new features in existing algorithms.

This project concerns algorithm design and analysis of efficient, highly accurate numerical methods for solving partial differential equations. Such equations are used in simulation of systems arising in diverse application fields such as aerospace engineering, semi-conductor device design, astrophysics, and biology. Even with today's fast computers, efficient computational solution of partial differential equations remains a challenge, and it is essential to design improved algorithms that can be used to obtain accurate solutions in these application models. The research aims to produce a suite of powerful computational tools suitable for computer simulations of the complicated solution structure in these applications. The project provides training for a graduate student through involvement in the research.

This project conducts research in algorithm development, analysis, and application of high order numerical methods, including discontinuous Galerkin (DG) finite element methods and finite difference and finite volume weighted essentially non-oscillatory (WENO) schemes, for solving linear and nonlinear convection-dominated partial differential equations, emphasizing scheme robustness, efficiency, and the treatment of stochastic effects. The project focuses on algorithm development and analysis. Topics of investigation include a new class of multi-resolution WENO schemes with increasingly higher order of accuracy, an inverse Lax-Wendroff procedure for high-order numerical boundary conditions for finite difference schemes on Cartesian meshes solving problems in general geometry, efficient and stable time-stepping techniques for DG schemes and other spatial discretizations, high order accurate bound-preserving schemes and applications, entropy stable DG methods, optimal convergence and superconvergence analysis of DG methods, numerical solutions of stochastic differential equations, and the study of modeling, analysis, and simulation for traffic flow and air pollution. Applications motivate the design of new algorithms or new features in existing algorithms; mathematical tools will be used to analyze these algorithms to give guidelines for their applicability and limitations and to enhance their accuracy, stability, and robustness; and collaborations with engineers and other applied scientists will enable the efficient application of these new algorithms.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.