## MAMCDTN-Numerical Methods

Module Provider: Mathematics and Statistics
Number of credits: 12 [6 ECTS credits]
Level:7
Terms in which taught: Autumn term module
Pre-requisites:
Non-modular pre-requisites:
Co-requisites: MAMCDTU Data & Uncertainty MAMCDTS Dynamical Systems MAMCDTE Partial Differential Equations
Modules excluded:
Current from: 2018/9

Module Convenor: Dr Hilary Weller

Type of module:

Summary module description:
Note: As this module is part of a joint programme with Imperial College London, the academic regulations for this module might differ from standard academic regulations usually applied at the University of Reading. The relevant document is the Joint Degree Programme Agreement between Imperial College London and University of Reading.

Aims:

Derive and implement finite difference schemes for solving first and second order linear PDEs

1. To introduce key concepts in the use of finite difference methods to discretise the fundamental terms in the equations of motion for atmosphere and ocean;

2. To describe advantages and disadvantages of different numerical techniques used in weather and climate models;

3. To use these topics to introduce general techniques for studying convergence and stability in numerical analysis;

4. To introduce key concepts in the theory of iterative methods for large linear systems arising from the numerical solution of PDEs, particularly motivated by the semi-implicit approach used in many state-of-the-art weather and ocean models;

5. To introduce state-of-the-art iterative methods that can be used on massively parallel computers;

6. To understand the challenges and possibilities for solving large-scale linear systems, and to pave the way for ?nonlinear problems in scientific computing.

Assessable learning outcomes:

• Apply and analyse a range of techniques in numerical theory for appropriate problems;

• Devise, analyse and apply a range of numerical techniques for partial differential equations;

• Apply and analyse a range of methods in numerical linear algebra;

• Implement a range of numerical methods on a computer.

Outline content:

• Finite difference methods for advection;

• Stability, boundedness, convergence of finite difference methods;

• Numerical methods for shallow water equations;

• Numerical conservation and dispersion;

• Lax-equivalence theorem, Domain of dependence, Von-Neumann stability analysis, Phase speed and dispersion errors, Conservation, Godunov's theorem;

• More advection schemes: Semi-Lagrangian, Artificial diffusion; The finite volume method; Lax-Wendroff and Warming and Beam, TVD schemes;

• Numerical methods for 2nd-order wave equations: Arakawa grids, Dispersion relations (outline of their derivation), Semi-implicit;

• How do coupled linear systems arise in weather and ocean models? Elliptic problems arising in non-hydrostatic ocean models, and implicit discretisations of wave equations; their infinite difference approximations;

• Gaussian elimination as repeated application of matrices, computational cost. Survey of when it works, pivoting. Quick discussion of round-off error;

• Classical iterative methods (Richardson, Jacobi, Gauss-Seidel, SOR) applied to infinite difference discretisation of Poisson's equation, convergence criteria by analysing the iteration matrix. Conditions on convergence for SOR parameter;

• Optimal value of for Richardson iteration. Convergence of Jacobi and Gauss-Seidel methods converge for diagonal dominant and irreducible diagonal dominant matrices. Optimal relaxation parameter in SOR for matrices that satisfy the “key property";

• Symmetric positive-definite matrices: convergence condition for Jacobi iteration and SOR. Optimal value of for SOR with symmetric matrices. SSOR algorithm, impact of symmetric iterative methods on iteration matrix;

• Termination criteria for iterative methods. Chebyshev acceleration; analysis using matrix polynomials;

• Conjugate gradient method; analysis using matrix polynomials, parallel implementation;

• GMRES; Krylov subspace, analysis using matrix polynomials, motivation for preconditioners;

• Criteria for good preconditioners. Additive Schwartz algorithm applied to finite difference approximation of Poisson's equation; convergence analysis;

• Brief sketch of multigrid methods.

Brief description of teaching and learning methods:

Lectures and tutorials.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 6 Guided independent study 94 Total hours by term 120.00 Total hours for module 120.00

Summative Assessment Methods:
 Method Percentage Set exercise 50 Class test administered by School 50

Summative assessment- Examinations:

Summative assessment- Coursework and in-class tests:

Formative assessment methods:
Peer marked tutorial questions.

Penalties for late submission:

Where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:
An average of 50% across the whole module.

Reassessment arrangements:
Via a written resit exam. Coursework will be carried forward if it received 40% or more, otherwise it must be resubmitted before the resit exam.