Scientific Computation and Numerical Analysis

Code

MATH3036

School

Mathematical Sciences

Level

3

Credits

20

Semesters

Spring UK

Summary

Differential equations play a crucial modelling role in many applications, such as fluid dynamics, electromagnetism, biomedicine, astrophysics and financial modelling. Typically, the equations under consideration are so complicated that their solution may not be determined by purely analytical techniques; instead one has to resort to computing numerical approximations to the unknown analytical solution. 

In this course we study numerical techniques for approximating data, ordinary and partial differential equations, and solving, or finding eigenvalues and eigenvectors of, the large linear systems of equations that result from these approximations. A detailed list of topics covered by this course is given below. 

Initial value problems (ODEs): multistage and multistep methods; convergence and stability; higher order ODEs; systems of first order ODEs; implicit methods. 

Boundary value problems (ODEs): finite differences for linear ODEs; error analysis; shooting method; eigenvalue problems. 

Partial differential equations: finite differences for elliptic, parabolic and hyperbolic PDEs; truncation error and stability analysis; finite volume methods. 

Approximation theory: least squares approximation; trigonometric polynomial approximation. 

Eigenvalues and eigenvectors:  power method; inverse iteration; Householder transformations; QR algorithm; singular value decomposition. 

Large linear systems: Krylov subspace methods; conjugate gradient method; preconditioning.

Target Students

This course is available to students studying within the School of Mathematical Sciences.

Classes

• Two 1-hour lectures each week for 12 weeks
• One 2-hour lecture each week for 12 weeks

Assessment

• 20% Coursework 1: Assessed piece of coursework involving the understanding, implementation and application of numerical algorithms outlined in the module within MATLAB.
• 20% Coursework 2: Assessed piece of coursework involving the understanding, implementation and application of numerical algorithms outlined in the module within MATLAB.
• 60% Exam 1 (3-hour): Written examination.

Assessed by end of spring semester

Educational Aims

Learning Outcomes

A student who completes this course successfully will be able to:

L1 - select, analyse and implement appropriate numerical methods for ordinary differential equations (initial and boundary value problems).

L2 - discretise partial differential equations using finite difference and finite volume methods.

L3 - analyse the discretisation error and stability of methods for approximating differential equations.

L4 - approximate functions and data using least squares and trigonometric polynomials.

L5 - approximate eigenvalues and eigenvectors using iterative algorithms.

L6 - formulate Krylov subspace methods for solving large linear systems of equations.

L7 - implement and evaluate numerical algorithms in Matlab.

Conveners

• Professor Kristoffer Van Der Zee