Applied and Computational Mathematics

The objective of this program is a training in modern mathematics aimed at applications and in the use of computers as a tool in the analysis, optimization, and control of physical and technological systems. It welcomes strong graduate students with a variety of backgrounds (ranging from the physical sciences and engineering to mathematics) wishing to work in partial differential equations and their applications. The program is sufficiently broad to accomodate software development and physical modelling as well as topics requiring delicate techniques in functional analysis or partial differential equations.

It is intended to offer students the possibility of collaborative contact with several local government and industrial research groups such as the Canadian Space Agency and a variety of other organisations with which members of the group have been involved at various times.

The program covers several closely connected areas which include:

There are no formal programmatic requirements beyond the departmental requirements. However the following guidelines should be followed and courses must be selected in consultation with an adviser from the group.

  1. All students should take courses in partial differential equations: appropriate courses are MATH 580 and MATH 581 at McGill and MAT 6110 at U de M.
  2. It is essential that most (and desirable that all) students develop their computational skills by taking appropriate courses in numerical analysis. Beyond the introductory courses, generally at an undergraduate level, the essential courses cover computational mathematics (MATH 578 at McGill and MAT6470 at U de M) numerical differential equations (MATH 579 at McGill) finite difference methods (MAT 6165 at U de M) and finite element methods (MTH 6206/7 at Polytechnique and MAT6450 at U de M).
  3. Students should develop an understanding of neighbouring areas of physics such as fluids and continuum mechanics, thermodynamics, etc. Suitable courses include MATH 555 at McGill and MAT 6150 at U de M; other useful courses can be found in Physics or Engineering departments.
  4. Students involved in fluid mechanics or material sciences should take a course on asymptotic and perturbation methods: MATH 651 at McGill or MTH 6506 at Polytechnique.
  5. Students in shape optimization or control should take at least one course in optimization. The following courses are available: MATH 560 at McGill, MAT 6428, MAT 6439 (Optimisation et contrôle), MAT 6441 (Analyse et optimisation de forme) at U de M; MTH 6403 and MTH 6408 at Polytechnique.
  6. Students who wish to work on shape optimization or the control of distributed parameter systems will need to develop a strong background in real analysis and functional analysis.

We expect that future elaboration and formalization of this program will occur within the framework described above which allows also for the introduction of additional areas under the broad umbrella of the program title.

For more information on our research programs and seminars, please consult the McGill Applied Mathematics page and the GIREF (Groupe interdisciplinaire de recherche éléments finis) page.

Courses 2009-2010

Automne/Fall

Optimization and Nonsmooth Analysis
Concordia MAST 681, Ron Stern
Introduction to nonsmooth analysis: generalized directional derivative, generalized gradient, normals and tangents. Nonsmooth calculus. Connections to convex analysis. Application to optimality conditions ans multipliers in nonlinear programming. Applications to dynamical systems (invariance issues). Thursday 13:15-14h30 LB 655
Numerical Analysis I
McGill MATH 578, Gantumur Tsogtgerel
Development, analysis and effective use of numerical methods to solve problems arising in applications. Topics include direct and iterative methods for the solution of linear equations (including preconditioning), eigenvalue problems, interpolation, approximation, quadrature, solution of nonlinear systems.
Applied Partial Differential Equations I
McGill MATH 580, Niky Kamran
Linear and nonlinear partial differential equations of applied mathematics. Uniqueness, regularity, well posedeness and classification for elliptic, parabolic and hyperbolic equations. Method of characteristics, conservation laws, shocks. Fundamental solutions, weak and strong maximum principles, representation formulae, Green's functions.
Calcul scientifique
Montréal MAT 6471, Anne Bourlioux et Robert G. Owens
Étude des algorithmes fondamentaux en calcul scientifique. Principes théoriques; programmation et application à des problèmes pratiques; utilisation scientifique de logiciels spécialisés.

Hiver/Winter

Topics in Applied Mathematics
Concordia MAST 680Q/MAST 865Q, L. Van Veen
Topic: "Projects in Dynamical Systems " The students will work individually on two projects in applied mathematics. Topics will be offered in the field of fluids dynamics, financial math or other applied dynamical systems. Rather then on theory, the emphasis will be on literature study, scientific writing, programming and working independently. Assesment will be based on written reports and an oral presentation.
Numerical Analysis of Nonlinear Equations
Concordia COMP 6361, Eusebius Doedel
An introduction to numerical algorithms for nonlinear equations, including discrete as well as continuous systems. The emphasis is on computer-aided numerical analysis, rather than numerical simulation. This course is suitable for scientists and engineers with a practical interest in nonlinear phenomena. Topics include computational aspects of homotopy and continuation methods, fixed points and stationary solutions, asymptotic stability, bifurcations, periodic solutions, transition to chaos, travelling wave solutions, discretization techniques. A variety of applications will be considered. Projects may be done on selected applications. Numerical software packages will be available.
Dynamical Systems
McGill MATH 574, Tony Humphries
Dynamical systems, phase space, limit sets. Review of linear systems. Stability. Liapunov functions. Stable manifold and Hartman-Grobman theorems. Local bifurcations, Hopf bifurcations, global bifurcations. Poincare Sections. Quadratic maps: chaos, symbolic dynamics, topological conjugacy. Sarkovskii's theorem, periodic doubling route to chaos. Smale Horseshoe.
Numerical Differential Equations
McGill MATH 579, Gantumur Tsogtgerel
Numerical solution of initial and boundary value problems in science and engineering: ordinary differential equations; partial differential equations of elliptic, parabolic and hyperbolic type. Topics include Runge Kutta and linear multistep methods, adaptivity, finite elements, finite differences, finite volumes, spectral methods.
Partial Differential Equations II
McGill MATH 581, Ming Mei
Systems of conservation laws and Riemann invariants. Cauchy- Kowalevskaya theorem, powers series solutions. Distributions and transforms. Weak solutions; introduction to Sobolev spaces with applications. Elliptic equations, Fredholm theory and spectra of elliptic operators. Second order parabolic and hyperbolic equations. Further advanced topics may be included.
Méthodes numériques pour les fluides
Montréal MAT 6151, Paul Arminjon
Equations scalaires et systèmes hyperboliques de lois de conservation. Solutions faibles. Raréfactions, ondes de choc, discontinuités de contact, unicité, conditions d'Entropie. Invariants et problèmes de Riemann. Equations d'Euler. Principales méthodes de Différences. Méthode de Godunov, Solveurs de Riemann : HLL, HLLC, Roe, Osher. Méthodes d'ordre 2 (Hancock-van Leer). Aperçu des Méthodes bidimensionnelles.
Références :
1) J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, 1983.
2) E. F. Toro , Riemann Solvers and Numerical Methodsfor Fluid Dynamics, Springer, 1997
3) R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.