Computational Mathematics III
Go to this course in the University Course Planner.
Description
In exploring large scale, complex systems, physicists, engineers, financiers and mathematicians often formulate problems as partial differential equations or many coupled ordinary differential equations. Only rarely can these mathematical models be solved algebraically. Instead computational mathematics derives approximate models that form the basis of computer predictions. Such models predict the climate, the weather, option prices, industrial processes, engineering devices, blood flow, epidemiology and more. This course develops sound stable computational methods for exploring large-scale systems.
Objective
To extend students' knowledge of the use of numerical (approximate) methods to solve problems involving algebraic, ordinary and partial differential equations and integrals. At the end of this subject students should be able to:
solve analytically intractable mathematical problems using numerical (approximate) techniques and assess the accuracy of the results; analyse the numerical stability of numerical techniques; write computer codes to implement numerical algorithms.
Content
Topics covered are: the numerical solution and stability of ordinary differential equations, using explicit and implicit methods; finite-difference and spectral methods applied to boundary value problems and certain partial differential equations, including Laplace's equation, the heat equation and the wave equation; stability analysis of these schemes; modern Krylov and multigrid methods are used to solve large systems of linear equations such as those that arise from finite-difference schemes; continuation methods.
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| Year |
Semester |
Level |
Units |
| 2012 |
1 |
3 |
3 |
Delivery
36 hours of lectures and tutorials.
Assessment
Ongoing assessment 30%, exam 70%.
Graduate attributes
Linkage past
Prerequisite is MATHS 1012 Mathematics IB (Pass Div I) or MATHS 2004 Mathematics IIM (Pass Div I). Assumed knowledge: APP MTH 2007 Differential Equations II or APP MTH 2000 Differential Equations and Fourier Series and a computer programming language (Matlab, Fortran or C).
Linkage present
This course contains material useful in many areas of applied mathematics including fluid mechanics, mathematical biology, industrial mathematics and mathematical finance. It includes techniques useful in other Level III Applied Mathematics courses such as APP MTH 3013 Differential Equations III, APP MTH 3002 Fluid Mechanics III and APP MTH 3006 Industrial Mathematics III. It is suitable for a student contemplating a career in industry, commerce, engineering or mathematical biology or medicine.
Linkage future
Level IV Applied Mathematics courses which may use the techniques considered in this course include APP MTH 4007 Computational Fluid Dynamics (Engineering), APP MTH 4003 Aerodynamics.
Restrictions
None.
Recommended text
Numerous books in the Barr-Smith Library can be found by checking under numerical analysis, numerical methods and computational mathematics.
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