Course Schedule
| Theory | Numerical | Exercises | Labs |
|---|---|---|---|
| Topology and Linear Algebra (Classes 1 and 2) | Functional Analysis and Fourier Theory (Classes 3 and 4) | Exercise Sheet 1 | Lab 1A: Getting familiar with Julia and its environment Lab 1B: The logistic map |
| –––––––––––- | –––––––- | –––––––- | ––––– |
| 1. Vectors, covectors, tensors, symmetries, complexification 2. Quotient spaces 3. Norms, induced norms 4. Inner product 5. Linear maps, invariant subspaces, eigenvalues-eigenvectors, exponentials, adjoint and unitary operators 6. Topological Spaces 7. Examples, Continuity, Compactness, Sequences, convergence 8. Stability of fixed points | 1. Basic Elements of Functional Analysis 2. Completing a normed space 3. Hilbert spaces 4. Riesz Representation Theorem 5. Sobolev spaces of positive integer indices and the Poincaré-Hardy theorem 6. Fourier Series and Sobolev Spaces 7. Basic properties of Fourier Series 8. Sobolev spaces of real indices, two important theorems | ||
| –––––– | –––––––- | –––––––- | ––––– |
| Ordinary Differential Equations - Analytical Studies (Classes 5 and 6) | Ordinary Differential Equations - Numerical Studies (Classes 7 and 8) | Exercise Sheet 2 | Lab 2A: Solving ordinary differential equations with numerical methods Lab 2B: Computing the stability region of some numerical integration schemes |
| 1. Definition, examples, uniqueness, existence 2. Reduction to first-order systems 3. Geometric interpretation 4. First integrals 5. Fundamental theorem, dependence on parameters, variational equation 6. Linear systems, general solution 7. Stability | 1. Defining the problem 2. Various approximation methods 3. Existence proof using Euler's method 4. Stability regions | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
| Evolutionary Partial Differential Equations - Analytical Studies (Classes 9 and 10) | Evolution Partial Differential Equations - Numerical Studies (Periodic Case) (Classes 11 and 12) | Exercise Sheet 3 | Lab 3A: Solving a single wave equation in a periodic domain Lab 3B: Solving a simple hyperbolic system |
| 1. Examples: advection and Burgers' equation 2. The Cauchy problem 3. Symmetric-hyperbolic systems: Wave equations, Maxwell's equations, Einstein's equations, etc. 4. Propagation cones 5. Existence and uniqueness, maximum propagation speed | 1. Method of lines 2. Discretizing space, finite differences 3. Discretizing time 4. Stability of evolution operators and the CFL condition | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
| Evolutionary Partial Differential Equations (Classes 13 and 14) | Evolution Partial Differential Equations - Numerical Studies (Boundary Conditions) (Classes 15 and 16) | Exercise Sheet 4 | Lab 4: Solving the second-order wave equation with boundaries and discontinuous interfaces |
| 1. The initial-boundary-value problem 2. Energy estimates with boundaries | 1. Operators satisfying summation by parts 2. Applying boundary conditions using penalty methods | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
| Non-linear Theory (Classes 17 and 18) | Other Evolutionary Equations (Classes 19 and 20) | Exercise Sheet 5 | Lab 5: Solving the heat equation |
| 1. An example (Burgers equation) 2. The general theory | 1. Parabolic equations (Heat equation) 2. Mixed systems (Navier-Stokes) 3. Schrödinger equation | ||
| Weak Solutions, Shocks (Classes 21 and 22) | Approximating Weak Solutions (Classes 23 and 24) | Exercise Sheet 6 | Lab 6: Solving Burgers equation |
| 1. Examples 2. Juncture conditions 3. Propagation Speeds 4. Lack of uniqueness 5. Entropy conditions | Lax-Friedrich and Weno algorithms for approximating weak solutions | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
| Stationary Partial Differential Equations - Analytical Studies (Classes 25 and 26) | Stationary Partial Differential Equations - Numerical Studies (Classes 27 and 28) | Exercise Sheet 7 | Lab 7: Solving the Laplacian with Dirichlet boundary conditions |
| 1. The boundary problem 2. Ellipticity 3. Example: the Laplacian 4. Weak existence and uniqueness 5. Generalizations | 1. Finite element theory 2. Solving problems in their weak formulation using Gridap | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
| Further Topics on Hyperbolic Systems (Classes 29 and 30) | Further Topics on Stationary Systems (Classes 31 and 32) | ||
| 1. Strongly Hyperbolic Systems 2. Constraints | 1. Lax convergence theorem 2. Finite element approximation theory 3. Variations, non-elliptic systems (min-max) | ||
| –––––––––––- | –––––––- | –––––––- | ––––– |
Last modified: November 16, 2024.
Website built with Franklin.jl and the Julia programming language.
Website built with Franklin.jl and the Julia programming language.