AMAT 31313: Computational Mathematics

Course Unit Code : AMAT 31313

Course Title           : Computational Mathematics

Pre-requisites         : AMAT 22292

Learning outcomes:

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

• classify Partial Differential Equations (PDE)

• identify initial and boundary conditions of PDE

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

• 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 schemes

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

• solve boundary value problems using basic finite elements methods

• solve one dimensional PDEs using finite element method by using appropriate software.

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 dimensional PDEs using finite element method: Weighted residual methods: Collocation method, least square method, Galerkin method.

Practical: 

• Simulate the Finite Difference solutions using appropriate programming language

Method of Teaching and Learning :  A combination of lectures, classroom discussions and computer laboratory sessions

Assessment : Based on assignments, group projects and Final examination.

Recommended Readings:

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

2. Smith, G. D. (3 rd 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. (2 nd 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.