Course: 2021/2022

Computational Techniques for Differential Equations

(18769)

Requirements (Subjects that are assumed to be known)

- Numerical methods at basic level.
- Knowledge of Mathematical Analysis in one and several variables.
- Knowledge of Linear Algebra.
- Knowledge of Ordinary Differential Equations and Partial Differential Equations.
- Basic knowledge of numerical methods for Ordinary Differential Equations.

One of the purposes of this course is to provide the basic techniques for the numerical resolution of EDPs. To do this end, we will analyze and establish the theoretical properties of each method (stability, precision, computational complexity) and we will demonstrate its operation with examples that describe its advantages and disadvantages. The main objective is to develop algorithmic thinking, emphasizing the main computational concepts.
More specifically, the aims of the course with respecty to the students include:
- Understanding of main numerical approximation methods for PDEs: finite difference method; finite element method; spectral methods for periodic and non-periodic problems.
- Ability to analyze the main features of a numerical method: order, stability, convergence.
- Ability to implement numerical methods for the solution of PDEs in one and two dimensions.
- Have criteria to assess and compare different methods depending on the problems to be solved, the computational cost and the presence of errors.
- Ability to program the algorithms studied in the course or use previously programmed algorithms (for example, in Matlab or Python).
CB6, CB7, CB8, CB9, CB10
CG1, CG2, CG3, CG4, CG5, CG6, CG7
CE1, CE2, CE3, CE4, CE5, CE6, CE8, CE9, CE10, CE11, CE12, CE13

Description of contents: programme

1. The Finite Difference Method
1.1 Introduction to Finite Difference Approximations
1.2 Steady States and Boundary Value Problems
1.3 Linear Elliptic Equations
1.4 Diffusion Equations and Parabolic Problems
1.5 Linear Advection Equations
2. The Finite Element Method in 1D
2.1 Piecewise Polynomial Approximation in 1D
2.2 The Finite Element Method in 1D
3. The Finite Element Method in 2D
3.1 Piecewise Polynomial Approximation in 2D
3.2 The Finite Element Method in 2D
4. Spectral Methods for Periodic Problems
4.1 Differentiation Matrices
4.2 Unbounded Grids: The Semi-Discrete Fourier Transform
4.3 Periodic Grids: The DFT and FFT
4.4 Smoothness and Spectral Accuracy
5. Spectral Methods for Non-Periodic Problems
5.1 Polynomial Interpolation and Clustered Grids
5.2 Chebyshev Differentiation Matrices
5.3 Boundary Value Problems
5.4 Chebyshev Series and the FFT
5.5 Time-dependent problems and stability regions

Learning activities and methodology

Each chapter provides examples, tests, exercises, and applications of the theory discussed. The course is based on well-established numerical methods. Students are expected to be able to write their own scripts or rewrite the scripts given by the professor.
There will be a theory session and a problem solving session of the selected exercises weekly. In addition to this, there will be 5 deliverable practices where students will have to implement the numerical methods discussed in class. The continuous evaluation will consist of these 5 practices (7% of the final mark each) plus a partial exam (25% of the final mark).
There will be a fixed office hours schedule (2h weekly) for the students.

Assessment System

- % end-of-term-examination 40
- % of continuous assessment (assigments, laboratory, practicals...) 60

Basic Bibliography

- Lloyd N. Trefethen. Spectral Methods in Matlab. SIAM. 2000
- Mats G. Larson and Fredrik Bengzon. The Finite Element Method: Theory, Implementation, and Applications. Springer. 2013
- Randall J. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM. 2007

Additional Bibliography

- C. Canuto, A. Quarteroni, M. Y. Hussaini and T. A. Zang. Spectral Methods: Fundamentals in Single Domains. Springer. 2006
- David Gottlieb and Steven A. Orszag. Numerical Analysis of Spectral Methods: Theory and Applications. SIAM. 1977
- G. D. Smith. Numerical Solution of Partial Differential Equations: Finite Difference Methods. Clarendon Press. 1985
- J. W. Thomas. Numerical Partial Differential Equations: Finite Difference Methods. Springer. 1995
- Jan S. Hesthaven, Sigal Gottlieb and David Gottlieb. Spectral Methods for Time-Dependent Problems. SIAM. 2007
- John C. Strikwerda. Finite Difference Schemes and Partial Differential Equations. SIAM. 2004
- Lloyd N. Trefethen. Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations . Cornell University. 1996
- Mark S. Gockenbach. Partial Differential Equations: Analytical and Numerical Methods. SIAM. 2011
- Susanne C. Brenner and L. Ridgway Scott. The Mathematical Theory of Finite Element Methods. Springer. 2008