Module Identifier MA34710
Module Title NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
Co-ordinator Professor Tim Phillips
Semester Semester 2
Other staff Professor Russell Davies
Pre-Requisite MA25110
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour example classes
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

#### Learning outcomes

On completion of this module, a student should be able to:
• discretise an elliptic partial differential equation using finite difference and finite elements methods.
• perform an error analysis for the discrete approximation to elliptic equations.
• solve the discrete equations using classical iterative methods.
• analyse the convergence behaviour of these iterative methods.
• discretise parabolic partial differential equations in one space variable.
• perform an error analysis for the discrete approximation to parabolic equations.

#### Brief description

Partial differential equations are the main means of providing mathematical models in science, engineering and other fields. Generally these models must be solved numerically. This course provides an introduction to numerical techniques for eliiptical and parabolic equations.

#### Aims

The aim of this course is to provide an introduction to numerical methods for solving partial differential equations of elliptic and parabolic type. Concepts such as consistency, convergence and stability of numerical methods will be discussed. Classical iterative methods for solving the systems of linear algebraic equations arising from the discretization of elliptic problems will be described and their convergence behaviour analysed.

#### Content

1. Finite difference approximations to elliptic partial differential equations. Local truncation error and error analysis. Boundary conditions on a curved boundary. Variational formulation and the finite element method. Classical iterative methods for solving linear systems of algebraic equations: Jacobi, Gauss-Seidel, SOR. Fourier analysis of convergence.
2. Finite difference approximations to parabolic partial differential equations in one space variable. Local truncation error and error analysis. Explicit and implicit methods. Convergence and stability. The Thomas algorithm.