AMAT 41373 : Advanced Computational Mathematics

Course Unit Code : AMAT 41373

Course Title           : Advanced Computational Mathematics

Pre-requisites         : AMAT 22292

 Learning outcomes:

Upon successful completion of the course unit the student will be able to:

• calculate finite difference operators to approximate the derivatives and corresponding truncation errors

• identify initial and boundary conditions of PDE

• apply finite difference methods to obtain the approximate solution of PDEs together with prescribed boundary and/or initial conditions

• analyze the stability, consistency and convergence of numerical scheme

• compare the accuracy of the approximate solution obtained by finite difference scheme with exact solution using simulation results

• solve boundary value problems using basic finite element methods

• solve one dimensional PDEs using finite element methods

• solve boundary value problems using free FEM++.

Course Content:

Finite Difference Methods: 

• Introduction, Classification of Partial Differential Equations (PDE): parabolic, hyperbolic and elliptic, Taylor series expansion: analysis of truncation error. Initial and boundary conditions: Dirichlet and Neumann boundary conditions. Finite difference methods: Forward, Backward, Centered and Crank-Nicholson schemes, Implicit and Explicit methods. Stability and Convergence analysis of numerical schemes: Von Neumann Analysis, Consistency and Stability, Lax Equivalent Theorem, Comparison of Numerical Schemes.

Finite Element Methods: 

• Introduction, Weak Formulation. Solving one and two dimensional PDEs using finite element method: Weighted residual methods: Collocation method, least square method, Galerkin method.

Practical: 

• Implement Finite Difference Schemes using appropriate software, Implement Finite element method using Free FEM++

Method of Teaching and Learning : A combination of lectures and tutorial discussions

Assessment : Based on tutorials, presentation, tests and end of course examination

Recommended Textbook:

1. Burden, R.L., Faires, J.D, Burden, M.L. (10th Ed., 2016). Numerical Analysis, Cengage Learning.

2. Smith, G.D. (3rd Ed., 1986). Numerical Solution of Partial Differential Equations: Finite Difference Methods, Clarendon press.

3. Evans, J., Blackledge, J. & Yardley, P. (2000). Numerical Methods for Partial Differential Equation, Springer.

4. Davies, A.J. (2nd Ed., 2011). Finite Element Method: An Introduction to Partial Differential Equations, OUP Oxford.

5. Desai, Y.M. (2011). Finite Element Method with Applications in Engineering, Pearson Education India.

6. Ŝolín, P. (2013). Partial Differential Equations and the Finite Element Method, Wiley.