Module Code - Title:

MS4008 - MATHEMATICAL METHODS 2: Numerical Methods for Partial Differential Equations

Year Last Offered:

2025/6

Hours Per Week:

Lecture

0

Lab

2

Tutorial

1

Other

0

Private

7

Credits

6

Grading Type:

N

Prerequisite Modules:

MS4404

Rationale and Purpose of the Module:

Having completed this module, the students should understand and be able to apply the standard finite difference methods for the numerical solution of two-dimensional linear partial differential equations; they should also understand how the finite element method is used to solve similar problems.

Syllabus:

Finite difference methods: Elliptic problems: stability, consistency and convergence; parabolic problems; explicit and implicit methods, Von Neumann stability analysis; hyperbolic problems; method of characteristics. Finite element method: Introduction to FEM for elliptic problems: analysis of Galerkin FEM for a model self-adjoin two point boundary value problem, weak solutions, linear basis functions, matrix assembly; extension of method to two dimensions, triangular and quadrilateral elements.

Learning Outcomes:

Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

* Given a finite difference scheme for an elliptic or parabolic differential equation, estimate the local truncation error. * Given a finite difference scheme for an elliptic differential equation, check whether it satisfies a discrete maximum principle; if it does, estimate the error of this numerical method. * Given a finite difference scheme for an initial value problem for a partial differential equation with constant coefficients, apply Von NeumannÆs method to find out whether this method is unconditionally stable, unconditionally unstable or conditionally stable. * Given a boundary value problem for an elliptic differential equation, obtain its variational formulation and functional minimization formulation; prove their equivalence for certain classes of solutions. * Find local and global stiffness matrices and load vectors for linear and quadratic finite elements applied to elliptic partial differential equations in one and two dimensions. * Given a finite difference or a finite element method for an elliptic or parabolic partial differential equation, implement this method into a MatLab code.

Affective (Attitudes and Values)

N/A

Psychomotor (Physical Skills)

N/A

How the Module will be Taught and what will be the Learning Experiences of the Students:

Mathematics lectures and tutorials in a computer laboratory. Students will have weekly homework (not for credit), a number of continuous assessment assignments overall worth 25%, and a final exam worth 75%.

Research Findings Incorporated in to the Syllabus (If Relevant):

Prime Texts:

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

Other Relevant Texts:

Becker, E.B., Carey, G.F. and Oden, J.D. (1981) Finite elements - an introduction, Vol. I , Prentice-Hall

Johnson, C. (1987) Numerical solution of partial differential equations by the finite element method , Cambridge University Press

Mitchell, A.R. and Griffiths, D.F. (1980) The finite difference method in partial differential equations , Wiley

Mitchell, A.R. and Wait R. (1977) The finite element method in partial differential equations , Wiley

Smith, G.D. (1985) Numerical solution of partial differential equations , Clarendon

Programme(s) in which this Module is Offered:

Semester(s) Module is Offered:

Module Leader:

natalia.kopteva@ul.ie