Degree | Type | Year | Semester |
---|---|---|---|
4313136 Modelling for Science and Engineering | OT | 0 | 2 |
Students should have basic knowledge of calculation, algebra, ordinary and partial differential equations, and basic notions of programming.
Partial differential equations allow deterministic mathematical formulations of phenomena in physics and engineering as well as biological processes among
many other scenarios. The objective of this course is to present the main results in the context of partial differential equations that allow learning about
these models and to study numerical methods for the approximation of their solution.
Introduction: General classification of partial differential equations, examples of models. Transport equation,
method of characteristics.
1. Parabolic equations
Fourier method. Heat equation. Fundamental solution, Gaussian kernel, convolution and solution formula for the pure initial value problem. Maximum principle and uniqueness of the solution. Numerical Methods: Finite difference methods for scalar parabolic equations: Euler Explicit, Euler Implicit and Crank-Nicolson methods: Von Neumann stability test. Parabolic stability CFL condition. Examples
2. Eliptic equations
Theory: Steady-state problems. Polar/Spherical coordinates: radial solutions. Dirichlet and Neumann boundary value problems. Poisson kernel. Applications. Euler-Lagrange equations associated to variational problems. Numerics and examples.
3. Hyperbolic equations
Scalar Conservation Laws. Weak solutions. Burgers equation. Shock waves and expansions fans. Hamilton-Jacobi equations and viscosity solutions. Introduction to the Level Set Method. Eikonal equation.
Numerical Methods: Finite difference methods in conservation form. Shock-capturing schemes. Monotone schemes: Lax-Friedrichs and upwind schemes. Convergence and stability conditions. Entropy-satisfying schemes. Examples. Level set method applications.
The objective of the classes of theory, problems and practices is to give to the students the most basic knowledge of the equations in partial derivatives
and their applications.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Classes of theory and problems | 30 | 1.2 | 5, 6, 10 |
Type: Supervised | |||
Internship classes | 8 | 0.32 | 11 |
Type: Autonomous | |||
Studies and practical work by the student. | 96 | 3.84 | 5, 6, 10 |
The assessment will consist of two partial exams and the delivery of the resolution of a problem through the computer.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First partial exam | 30% | 4 | 0.16 | 2, 1, 4, 3, 8, 7, 5, 6, 9, 10, 11 |
Second partial exam | 30% | 4 | 0.16 | 10 |
Solution of a problem with a computer | 40% | 8 | 0.32 | 2, 1, 4, 3, 8, 7, 5, 6, 9, 10 |
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19 (2nd ed.), Providence, R.I., American Mathematical Society, (2010).
B. Gustafson, H-O. Kreiss and J. Oliger, Time dependent problems and Difference Methods, Wiley-Intersciences, (1996).
F. John, Partial Differential equations, vol. 1, Applied Math Sciences, Springer, (1978).
P.D. Lax, Hyperbolic systems of Conservation Laws and The Mathematical Theory of Shock Waves SIAM, 1973.
R.J. LeVeque, Finite Volume Methods for Hyperbolic problems, Cambridge University Press, 2002.
Y.Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge 2005.
S. Salsa, Partial differential equations in action : from modelling to theory Springer, 2008.
G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, (1986).
E.F. Toro Riemann Solvers and Numerical Methods for Fluid Dynamics: A practical Introduction, Springer-Verlag, 2009.
G.B. Whitham Linear and nonlinear Waves, Wiley-Intersciences, (1999).
We leave full freedom to students to use the language that suits them best to do the numerical exercises of this subject.