Module MATH4221: NUMERICAL DIFFERENTIAL EQUATIONS IV

Department: Mathematical Sciences

MATH4221: NUMERICAL DIFFERENTIAL EQUATIONS IV

Prerequisites

• Mathematics modules to the value of 100 credits at Years 2 and 3, with at least 40 credits at Level 3, and including Numerical Analysis II (MATH2051).

Corequisites

• None.

Excluded Combination of Modules

• Numerical Differential Equations III (MATH3081)

Aims

• To develop a basic understanding of the theory and methods of numerical solution for differential equations, both ordinary and partial.

Content

• Introduction to numerical methods for initial-value problems.
• Local and global truncation errors, convergence.
• One-step methods, with emphasis on explicit Runge-Kutta methods.
• Linear multistep methods, in particular the Adams methods
• Predictor-corrector methods and estimation of local truncation error.
• Stability concepts and stiff problems.
• Finite difference methods for parabolic PDEs, CFL-condition, stability.
• Finite element or finite difference method for elliptic equations.
• Iterative methods to solve linear systems.
• Eigenvalue problem for elliptic operators.
• Finite volume or finite difference method for the wave equation, CFL condition.
• Specialist software will be used for computational work throughout the module.
• Reading material on one or more aspects of implicit Runge-Kutta methods.

Learning Outcomes

• By the end of the module students will: be able to solve novel and/or complex problems in Approximation Theory and Solution of Ordinary Differential Equations.
• have a systematic and coherent understanding of theoretical mathematics in the field of Approximation Theory and Solution of Ordinary Differential Equations.
• have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Consistency, convergence and linear stability of numerical methods.
• One-step methods, particularly Runge-Kutta methods.
• Linear multi-step methods.
• Predictor-corrector methods
• Finite difference methods for parabolic and elliptic PDEs
• Iterative methods to solve linear systems.
• Eigenvalue problem for elliptic operators.
• Knowledge and understanding of a topic in the areas listed under content, for implicit Runge-Kutta methods.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, Computation.
• Ability to read independently to acquire knowledge and understanding in the area of implicit Runge-Kutta methods.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Subject material assigned for independent study develops the ability to acquire knowledge and understanding without dependence on lectures.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
• The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned in each of Michaelmas and Epiphany terms

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University