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Department of Mathematics



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MA3011 Applied Numerical Mathematics

MA3011 Applied Numerical Mathematics

Credits: 20 Convenor: Professor B. Leimkuhler Semester: 1
Prerequisites: essential: MA1001(=MC126), MA1002(=MC127), MA1151(=MC146), MA1152(=MC147), MA2001(=MC224), MA2101(=MC248), MA2102(=MC241), MA2071(=MC228) desirable:
Assessment: Regular coursework and computer assignments, computer project (written report, oral presentation): 50% 1.5 hour exam: 50%
Lectures: 18 Problem Classes: none
Tutorials: none Private Study: 104
Labs: 18 Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

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To be able to identify certain types of elliptic, parabolic and hyperbolic partial differential equation;
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To be able to classify various types of ordinary differential equations such as stiff systems and Hamiltonian systems;
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To know physical contexts in which these differential equations may arise and the types of boundary conditions needed for well-posedness.
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To be able to construct finite difference schemes for differential equations, and, where appropriate, analyze their stability;
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To know and understand appropriate linear algebra methods needed in the implementation of these schemes;
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To be able to implement the schemes in Matlab.

Subject Skills

Aims

To equip the student with some of the advanced numerical methods and programming skills required in engineering and research laboratories.

Learning Outcomes

Programming skills; the ability to solve differential equations and large linear algebra problems numerically. The course will include the design of a substantial computer project, a written project report, and the oral presentation of such a project.

Assessment

The coursework will consist of regularly assigned exercise sheets, including computer assignments. There will be weekly instructor-assisted lab sessions. A substantial individual computer project will be required and the student will be required to present the project orally to the class and the instructor.

The June examination will have 4 questions, it being (theoretically) possible to obtain full marks by answering 3 of them.

Explanation of Pre-requisites

Students must be familiar with elementary material on ordinary differential equations (MA1002) and multivariate calculus (MA1001, MA2001). A basic working knowledge of linear algebra, and an introductory course in scientific computing (MA2071) will be required, as will a working knowledge of classical real analysis (MA2101). Students must be able to program in MATLAB (MA2071).

Course Description

This course will introduce students numerical approaches to scientific problems, including applications in engineering and physics. Various nonlinear differential equations will be introduced as simplified models for real-world systems, discretizations will be introduced, numerical methods will be developed for each application, and computer implementation (primarily in Matlab) will be considered. The idea is to help students learn how to tackle an application using numerical methods and to relate the computational results obtained to the model setting.

Examples of problems to be studied include elliptic systems governing the load on an elastic membrane, the parabolic heat equation for temperature distribution in a bar, and nonlinear hyperbolic wave equations. Methods to be studied include iterative methods such as conjugate gradients for solving elliptic differential equations, stiff ordinary differential equation solvers for solving the heat equation, and symplectic discretizations for conservative models.

The emphasis will be on practical aspects of each model and implementation of numerical methods, but some theory will be also be developed.

Syllabus

The course will introduce (a) boundary value problems for elliptic partial differential equations in two dimensions, (b) initial/boundary value problems for parabolic partial differential equations in one and two dimensions, (c) hyperbolic nonlinear partial differential equations, especially the nonlinear wave equation. It will introduce finite difference schemes appropriate to these problems, including standard centred finite differences for elliptic equations, Crank-Nicholson and related methods. Stability analysis by evolution of Fouries modes will be presented, together with the Courant-Friedrichs-Lewy (CFL) condition for stability.

Finite difference schemes replace partial differential equations by large systems of linear equations. Numerical linear algebra methods to be presented will include SOR (for simple problems) and Krylov subspace methods.

Ordinary differential equation solvers including elementary schemes such as Trapezoidal Rule, Euler's Method, and explicit Runge-Kutta methods will also be presented and analysed. A comparison of stiff and non-stiff ODEs and appropriate numerical methods will be given. The Leapfrog integrator will be introduced for solving conservative semi-discretized nonlinear wave equations.

A major component of the course will be the development of a substantial computer project to tackle an application. Students will present this work both in written form and in a 20-minute oral presentation.

Reading list

Recommended:

D.F. Mayers and K.W. Morton, Numerical solution of partial differential equations: an introduction, Cambridge University Press (1994).

Recommended:

G. Golub and J. Ortega, Scientific Computing and Differential Equations : An Introduction to Numerical Methods, Academic Press (1991).

Resources

Marked problem sheets, computer assignments.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA3061 Ordinary Differential Equations Up: ModuleGuide03-04 Previous: MA3001 Relativity and Electromagnetism

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Last updated: 2004-02-21
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