Numerical Solution of Partial Differential Equations

Cod y Modiwl
MA33520
Teitl y Modiwl
Numerical Solution of Partial Differential Equations
Blwyddyn Academaidd
2026/2027
Semester
Semester 2
Cyd-gysylltydd y Modiwl
Dr Daniel Peck
Rhestr Ddarllen
Gweld ar Aspire
Rhagofynion
MA25220 neu MA34110 neu MT25220 neu MT34110
Staff Eraill sy'n Cyfrannu

Dulliau Asesu

Math o Asesiad

Manylion Asesiad

Hyd Asesiad

Cyfran
Asesiad Semester Group Report: On numerical implementation of the finite difference method 2000 Words 25%
Asesiad Semester Individual Report: On numerical implementation of the finite element method 2000 Words 35%
Arholiad Semester Assessed Practical: Exam in computer rooms. 2 Awr 40%
Asesiad Ailsefyll Individual Report: Individual report on the finite difference/finite element methods. 2000 Words 60%
Arholiad Ailsefyll Assessed Practical: Exam in computer rooms. 2 Awr 40%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

  1. Make an appropriate choice of numerical method (finite difference, finite element, hybrid) for a given linear elliptic, parabolic or hyperbolic PDE.
  2. Discretise partial differential equations of arbitrary order using finite difference methods.
  3. Use analytical techniques to determine the stability criterion for, and demonstrate consistency of, a finite difference scheme for a second order linear PDE and determine the local truncation error.
  4. Derive the variational formulation for a given boundary value problem, and obtain the associated finite element discretisation.
  5. Determine an appropriate set of shape functions with which to obtain the finite element approximation.
  6. Implement a finite difference scheme in a Python environment to solve a two-dimensional hyperbolic PDE with boundary conditions.
  7. Implement the finite element method in a Python environment to solve a one-dimensional boundary value problem.
  8. Perform error and convergence analysis on finite difference and finite element schemes in a Python environment.

Disgrifiad cryno

This module provides an introduction to numerical methods for solving partial differential equations of elliptic, parabolic and hyperbolic type. The analytical basis and numerical implementation of both finite difference and finite element methods are covered. Concepts such as consistency, convergence and stability of numerical methods will be discussed.

Cynnwys

Finite difference approximations of hyperbolic and one-dimensional parabolic partial differential equations. Convergence of a finite difference scheme: consistency, well-posedness and stability. Local truncation error and error analysis. Numerical implementation of two-dimensional problems in a python environment.

Variational formulation, shape functions and the finite element method for a one-dimensional problem. Numerical implementation in a Python environment. Extension of the method to two dimension elliptic problems. Bounds on the solution error.

Formulation of hybrid finite difference/finite element methods for parabolic (evolution) problems.

Suitability of the finite difference, finite element or hybrid methods to approximate given elliptic, parabolic and hyperbolic partial differential equations.

Sgiliau Modiwl

Math o Sgiliau

Manylion Sgiliau
Addasrwydd a gwydnwch Students are expected to develop their own approach to time-management and to use the feedback from marked work to support their learning.
Cydlynu ag erail Students will work together to complete the finite difference report.
Cyfathrebu proffesiynol Writing of reports.
Datrys Problemau Creadigol Applied to numerical methods.
Gallu digidol Implementing the numerical schemes in Python.
Sgiliau Pwnc-benodol Broadens exposure of students to topics in mathematics, and an area of application that they have not previously encountered.

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 6