Module MATH3081: NUMERICAL DIFFERENTIAL EQUATIONS III

Department: Mathematical Sciences

MATH3081: NUMERICAL DIFFERENTIAL EQUATIONS III

Prerequisites

• Numerical Analysis II (MATH2051) AND Analysis in Many Variables II (MATH2031).

Corequisites

• One 20 credit Level 3 mathematics module.

Excluded Combination of Modules

• Numerical Differential Equations IV (MATH4221)

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.

Learning Outcomes

• By the end of the module students will: be able to solve novel and/or complex problems in Numerical Differential Equations.
• have a systematic and coherent understanding of theoretical mathematics in the field of Numerical 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.

• In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, Computation.

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.
• 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:

Eight 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