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MCSC 6000G Graduate Seminar in Modelling and Computation Science. This course is a year-long Seminar Series on Modelling and Computational Science which will take place weekly for the entire academic year. Every graduate student enrolled in this course must give a presentation on a research topic. In addition to the student presentations, the seminar will feature speakers from UOIT and invited speakers from academia, industry and government. Successful completion of the course will also require attendance at the UOIT Faculty of Science Colloquium Series. 0 cr, 1 lec. Prerequisite: Successful completion of all Core Courses in the Program. 

MCSC 6001G Thesis.

MCSC 6002G Research Project.

MCSC 6010G Mathematical Modelling. This is a core course and forms an essential part of the MSc program. The student will get familiar with the fundamental principles and techniques in mathematical modelling, showcased through the use of classical and advanced models in physics, biology and chemistry. Several analytical techniques will be introduced through the study of the mathematical models presented. Topics may include: Population models and epidemiology, neuron and cell dynamics, nonlinear waves in biological, chemical and physical systems, fluid dynamics, pattern formation (in fluid experiments, animal coat patterns, chemical reactions, visual cortex), coupled systems (neurons, traffic flow, lattice systems). 3 cr, 3 lec. Prerequisite: Admission to the MSc program in Modelling and Computational Science. 

MCSC 6020G Numerical Analysis. Numerical analysis is the study of computer algorithms developed to solve the problems of continuous mathematics. Students taking this course gain a foundation in approximation theory, functional analysis, and numerical linear algebra from which the practical algorithms of scientific computing are derived. A major goal of this course is to develop skills in analysing numerical algorithms in terms of their accuracy, stability, and computational complexity. Topics include best approximations, least squares problems (continuous, discrete, and weighted), eigenvalue problems, and iterative methods for systems of linear and nonlinear equations. Demonstrations and programming assignments are used to encourage the use of available software tools for the solution of modelling problems that arise in physical, biological, economic, or engineering applications. 3 cr, 3 lec. Prerequisite: Admission to the MSc. Program in Modelling and Computational Science.  

MCSC 6030G High-Performance Computing. The goal of this course is to introduce students to the tools and methods of high-performance computing (HPC) using state-of-the art technologies. The course includes an overview of high-performance scientific computing architectures (interconnection networks, processor arrays, multiprocessors, shared and distributed memory, etc.) and examples of applications that require HPC. The emphasis is on giving students practical skills needed to exploit distributed and parallel computing hardware for maximizing efficiency and performance. Building on MCSC 6020G, students will implement numerical algorithms that can be scaled up for large systems of linear or nonlinear equations. Topics may include: survey of computer architectures; efficiency guidelines for HPC; parallel algorithm design; programming tools; timing, profiling, and benchmarking; optimizations. 3 cr, 3 lec. Prerequisite: MCSC 6020G Numerical Analysis or equivalent. 

MCSC 6120G - Numerical Methods for Ordinary Differential Equations. Differential equations are an indispensable tool for the modelling of physical phenomena. However, most often in practice, analytical solutions to model equations cannot be found, and numerical approximations must be made. In this course, practical computational techniques for the numerical solution of ordinary differential equations will be covered, with an emphasis on their implementation and the fundamental concepts in their analysis. Topics include: Numerical methods for initial value problems: forward and backward Euler and trapezoidal scheme; implicit and explicit Runge-Kutta methods, including general formulation; Linear multistep methods: Adams-Bashforth, Adams-Moulton, Backward Differentiation Formulae (BDF); Numerical methods for boundary value problems: simple and multiple shooting and difference schemes. In association with the techniques, topics such as convergence, accuracy, consistency, 0-stability, absolute stability, A-stability, stiffness, and error estimation and control will be discussed. 3 cr, 3 lec. Prerequisite: Numerical Analysis MCSC6020G or equivalent. 

MCSC 6125G Numerical Methods for Partial Differential Equations. Partial differential equations (PDEs) constitute a vital modelling tool on science and a rich field of mathematical research. This course is an introduction to the mathematical concepts required to develop accurate, reliable, and efficient numerical software for the approximate solution of PDEs. Essential model problems of elliptic, parabolic, and hyperbolic type are examined with corresponding numerical approximation techniques. Approximation schemes are compared and contrasted with an emphasis on the convenience of available software as well as error estimation, consistency, stability, and convergence. Topics may include: finite-difference, finite-element, finite-volume, and spectral collocation methods; Von Neumann analysis; time-stepping algorithms and the method of lines; dissipation and dispersion; error estimates; iterative methods. 3 cr, 3 lec. Prerequisite: Numerical Analysis MCSC 6020G. 

MCSC 6140G Dynamical Systems and Bifurcations. This course provides an introduction to the modern theory of dynamical systems and bifurcation theory, including chaos theory. Dynamical systems theory is an important tool in the modelling of many physical systems, but it is also a rich field of mathematical research in itself. By the end of this course, the student will have acquired a large toolkit of techniques to analyse the dynamical features of ordinary differential equations and discrete dynamical systems. Topics include: Structural stability, invariant manifolds, local and global bifurcations, reduction methods, routes to chaos, applications. 3 cr, 3 lec. Prerequisite: Undergraduate modern theory of ordinary differential equations. 

MCSC 6150G Fluid Dynamics. This course will give a unified view of fluid dynamics by emphasizing mathematical structures that reappear in different guises in almost all sub-fields of fluids. The student will become familiar with the fundamental principles, techniques and basic equations in fluid dynamics and will come to appreciate the basic nonlinear nature of most fluid flows. Topics include: Reynolds number and other non-dimensional parameters, stability and scaling, turbulence and the transition from laminar flow to turbulence, Newtonian and non-Newtonian flows, eigenmodes of a flow problem, including nonlinear exchange of energy between modes, lattice-gas and Boltzmann models. 3 cr, 3 lec. Prerequisite: Admission to the MSc program in Modelling and Computational Science. 

MCSC 6160G Transport Theory. The course is a general introduction to transport theory. Continuous-medium transport and discrete-particle transport are presented in a unified manner through the use of the probability distribution function. Various types of transport problems are presented together with analytic solutions for the simpler problems that allow them. Approximate and numerical methods are also covered. Topics include: Particle distribution functions, generic form of transport equation, particle streaming, one-speed transport theory, linear collision operators, the Boltzmann collision term, diffusion theory, hydrodynamic equations, eigenvalue problems, boundary value problems, perturbation and variational approximation methods, deterministic numerical methods, Monte Carlo simulations. 3 cr, 3 lec. Prerequisites: Linear algebra, differential equations, vector calculus. 

MCSC 6165G Monte Carlo Methods. This course provides an introduction to the simulation of stochastic processes using Monte Carlo methods. Concepts presented will include pseudo-random number and random variate generation, Markov chain models, Monte Carlo integration, variance reduction, and numerical optimization. Applications may include: solution to the Boltzmann transport equation (specifically for radiation transport) statistical physics, biophysics, and queuing theory. 3 cr, 3 lec. Prerequisites: Undergraduate-level theory of ordinary and partial differential equations, and introductory statistics. 

MCSC 6170G Computational Chemistry. Accessible introduction to the fundamental principles underlying different methods from classical to quantum theories, and from first principles through to the latest advances in the area. The main focus is on molecular structures and energetics. Molecular properties and aspects of spectroscopy and dynamics are also covered. Topics include: force-field and electronic-structure methods, electron correlation, basis sets, density functional theory, relativistic methods, hybrid quantal/classical models, excited electronic states, wavefunction analysis, molecular properties, transition state theory and reaction dynamics, optimization techniques. 3 cr, 3 lec. Pre-requisites: Introductory quantum mechanics and undergraduate mathematics. MCSC 6010G, MCSC 6020G. 

MCSC 6180G Computational Physics. The course introduces the fundamental principles which form the basis for carrying out modern HPC simulations in physics, chemistry and materials science, their realization in the form of various numerical algorithms, and applications to different problems and real-world systems. The main focus is the study of advanced methods of studying quantum-mechanical and statistical mechanical systems. Approaches considered will include: density functional theory (DFT) and its formulation in terms of pseudopotential and all-electron methods (DFT will be extended to treat excited states and, in particular, the optical properties of materials); molecular dynamics simulation, which will be used to describe ground-state properties such as atomic structure, vibrations and phase transitions, and the structural properties of fluids and fluid mixtures; and Monte Carlo simulation, which will used to provide molecular-level descriptions of various materials, fluids and fluid mixtures. 3 cr, 3 lec. Prerequisites: Undergraduate-level quantum mechanics and statistical mechanics. 

MCSC 6210G -Advanced Topics in Mathematical Modelling. This course builds on the core course Mathematical Modelling and elaborates on some of its topics in greater detail. In addition, it introduces a variety of special topics in applied mathematics with a focus on industrial and natural processes and phenomena. The topics will be chosen according to the needs and demands of the students and the available faculty resources. Topics and application may include: Auto-correlation of data sets, bifurcations in time-series, embedding time series, modelling stochastic systems, perturbation methods for partial differential equations, travelling wave phase plane, advanced reaction-diffusion phenomena and transition layers, Hausdorff measures, fractal dimension, Belousov-Zhabotinsky reaction, analysis of heartbeat time-series, fractals in science and medicine, chaotic dynamics in symmetric coupled cell systems, time series in the stock market and other financial products. 3 cr, 3 lec. Prerequisite: MCSC 6010G.  

MCSC 6220G Advanced Topics in Numerical Analysis. This course explores recent problems in numerical analysis that are at the forefront of current research. The main objective of the course is to familiarize students with contemporary theoretical results and practical algorithms as preparation for doctoral research. The topics will be chosen according to the needs and demands of the students. Potential topics include: level-set methods, finite element methods, finite volume methods, spectral methods, numerical optimization, multigrid methods, numerical linear algebra, Krylov subspace methods, preconditioning iterative methods.. 3 cr, 3 lec. Prerequisite: MCSC 6020G. 

MCSC 6230G Advanced Topics in High-Performance Computing. This course explores recent topics in high-performance computing that are at the forefront of current research. The main objective of the course is to familiarize students with contemporary implementations and practical algorithms as preparation for doctoral research. The topics will be chosen according to the needs and demands of the students. Potential topics include: distributed computing, cluster computing, grid computing, numerical linear algebra for high-performance computers, domain decomposition methods, parallel preconditioners. 3 cr, 3 lec. Prerequisite: MCSC 6030G. 

MCSC 6240G Advanced Topics in Dynamical Systems. This course builds on the topics covered in MCSC 6140G. The course covers advanced material, including recently developed tools, for the analysis of dynamical systems. By the end of the course, the student will be able to perform a bifurcation analysis of models that they will encounter in research or industry, including judging when such analysis is appropriate, choosing the right tools and interpreting the results. The topics will be chosen according to the needs and demands of the students. Potential topics include: equivariant bifurcation theory and applications, bifurcations in delay and partial differential equations, numerical continuation of bifurcations for ordinary, delay and partial differential equations, bursting in biological phenomena and other systems. 3 cr, 3 lec. Prerequisites: MCSC 6140G, MCSC 6020G. 

MCSC 6280G Advanced Topics in Computational Science. This course explores recent problems in Computational Science that are at the forefront of today's research. The main objective of the program is to bring students up to date with the current state of the art of Computational Science and make them ready for Ph.D research. The topics will be chosen according to the needs and demands of the students and the availability of faculty. Potential topics include: computational cluster science, quantum computing: concepts, advantages and problems, quantum Monte Carlo: applications in computational physics, advanced molecular simulations, advanced optimization, advanced Monte Carlo simulations. 3 cr, 3 lec. Prerequisites: MCSC 6010G, MCSC 6020G. Co-requisites: As required by the subject matter; e.g., MCSC 6170G Computational Chemistry, MCSC 6180 Computational Physics, MCSC 6165G Monte Carlo Methods.

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