Computational Modelling

Complex physical phenomena, from flows to structural deformations, are often modelled using partial differential equations (PDEs). This course provides an introduction to the numerical solution of PDEs . We begin by describing the general principles of computational modelling, and investigate the properties of PDEs derived from common physical problems. We then describe two popular discretisation approaches, the finite-difference and finite-element methods, along with techniques used for their analysis and practical application. In the last part of the course, we consider time-march methods for unsteady problems, and basic iterative techniques for the solution of large algebraic systems.

1. Introduction to computational modelling

- Errors in the numerical representation of physical phenomena

- Requirements for discretisation methods

2. Classification and boundary conditions

- A review of model PDEs

- Basic classification of PDEs

- Definition of characteristics, Elliptic, hyperbolic and parabolic PDEs

- Dirichlet and Neumann boundary conditions, well-posed problems

3. Finite-difference methods

- Modified equation, Taylor table

- Upwinding and artificial dissipation

- Generalised transformation

- Fourier analysis

4. Verification

- Method of manufactured solutions

- Code and solution verification, Richardson extrapolation.

5. Spectral and finite-element methods

- Method of weighted residuals

- Element and global assembly

- Application of boundary conditions

- Unsteady problems

6. Analysis of time-march methods

- Accuracy of transient computations

- Systems of ODEs, semi-discrete eigenvalues

- Wave space, relation to Fourier analysis

- Fully-discrete eigenvalues, stability, stiffness

7. Iterative solution of algebraic systems

- Direct solution methods versus iterative solution methods

- Jacobi and Gauss-Seidel techniques

- Convergence rates, stopping criteria