MA4004 Numerical Solution of Differential Equations

This is the former 423/MA4004/MT4421

 SEMESTER: First
CONTACT: Dr Joel Daou (M/O9) CREDIT RATING: 15
Aims:
  • To provide a unified account of numerical methods for solving differential equations.
  • To emphasise the need for careful analysis and concepts, together with efficient implementation in practice.
  • To stress the importance of the whole subject in practical problems arising in engineering and the physical sciences
Intended Learning Outcomes: On successful completion of the course unit students will be able to 
  • solve differential equations using efficient numerical methods; 
  • Understand error and convergence concepts for numerical methods;
  • Carry out stability investigations and appreciate their relevance to practical computation.
Pre-requisites: 157, 211, 215, 362 (ex-UMIST)
Dependent Courses: MT4882
Course Description: A high level introduction to numerical methods for solving ordinary and partial differential equations.  For systems of ODEs the emphasis is on stiff problems.  This involves stability investigations for problems and methods.  Practical implementation details and representations/control strategies of well-known codes are also covered.  Finite difference methods are introduced in the setting of two-point boundary-value problems for ODEs.  Finally we deal with the use and analysis of finite difference methods for PDEs.  This is done by studying well-known equations of stationary and evolutionary type.
Teaching Mode: 2 Lectures per week
1 tutorial per week
Private Study: 5 hours per week
Recommended Texts: U M Ascher, R M M Mattheij and R D Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, 1995, SIAM.
E Hairer, S P Norsett and G Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, (2nd edition), 1993, Springer-Verlag.
E Hairer and G Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, (2nd edition), 1996, Springer-Verlag.
A Iserles, Numerical Analysis of Differential Equations, 1996, Cambridge University Press.
K W Morton and D F Mayers, Numerical Solution of Partial Differential Equations, 1994, Cambridge University Press.
Assessment Methods: Coursework: 25%
Coursework Mode: One assignment, deadline in Week 10.
Examination: 75%
For MMath students, the examination is of 2 hours duration at the end of the First Semester.
For MSc students, the examination is of one and a half hours duration at the end of the First Semester.
No. of Lectures Syllabus
4 Discrete methods for solving ODEs: Review of Runge-Kutta methods, linear multistep methods (Adams) convergence and order. A number of practical ODE/PDE problems from different areas of applications will be introduced. They will be used and solved to illustrate ideas throughout the course.
4 Stiff systems: Discussion and definitions of stiffness. Stability analysis based on test problems. Implicit Runge-Kutta methods. Stability regions, A-stability and other stability concepts. The BDF methods.
3 Implementation of methods: Error control and efficiency questions. Variable stepsize and variable order. Iterative solution of the implicit equations in stiff solvers. Discussion of some well-known codes.
3 Boundary value problems: Shooting methods for two-point boundary value problems, continuation. Finite difference methods for linear equations and for more general problems. Implicit Runge-Kutta methods, deferred correction.
4 Finite difference methods for Poisson's equation: The 5-point formula. Existence and order of convergence for the grid-solution. Curved boundaries and derivative boundary conditions. Higher order difference schemes.
6 Evolutionary equations: The heat conduction equation. A simple explicit finite difference scheme. Stability and convergence. Semi-discretisation (the method of lines), stiffness. The advection equation. Finite difference schemes, Lax-Wendroff, leapfrog. The CFL condition and the Von Neumann stability test.

 Last revised August, 2006