# Class schedule

*Note*: This schedule is subject to revision during the term.
**Monday, January 4**

IVP for ODE. The Picard-Lindelöf theorem. Lipschitz condition. Stability. Euler's method.
**Wednesday, January 6**

One-step methods. Global error analysis. Backward Euler.
**Monday, January 11**

RK family. Order conditions. Adaptive methods. Embedded RK pairs.
**Wednesday, January 13**

LMM. Consistency. Order conditions. Zero-stability. Root condition. Convergence.
**Monday, January 18**

Dahlquist first barrier. Derivation of LMM. PECE. Milne device.
**Wednesday, January 20** - (*Homework 1 due*)

Stiff problems. Absolute (or linear) stability. Stability region. A-stability.
**Monday, January 25**

Stability region of LMM. Root locus curve. Dahlquist second barrier. A(α)-stability. L-stability.
**Wednesday, January 27**

Stability function of RK methods. Collocation.
**Monday, February 1**

Gauss, Radau and Lobatto methods. Implementation of IRK. One-sided Lipschitz condition. Phase space flow.
**Wednesday, February 3** - (*Homework 2 due*)

Monotonicity. B-stability and algebraic stability. Numerical flow. Linear and quadratic invariants.
**Monday, February 8**

Gradient flows. Conservative systems. Hamiltonian systems. N-body problem. Energy and momentum. Angular momentum. Partitioned RK. Reversibility.
**Wednesday, February 10**

Adjoint. Symmetric methods. Composition. Reversible methods. Symplectic flows.
**Monday, February 15**

Two point BVP. Shooting. Finite difference methods.
**Wednesday, February 17** - (*Homework 3 due*)

Stability analysis of finite difference methods. Discrete Green's function.
**February 22–26** - Study week

**Monday, March 1**

Stability in 2-norm. Conditioning of the discretized system.
**Wednesday, March 3** - (*Homework 4 due*)

Collocation and Petrov-Galerkin frameworks. Fourier series.
**Monday, March 8**

Convergence of Fourier series.
**Wednesday, March 10**

Fourier-Galerkin methods. Trapezoidal quadrature and DFT. FFT. Chebyshev polynomials.
**Monday, March 15**

Multidimensional BVP. Weak formulation. Finite elements. Element-by-element assembly.
**Wednesday, March 17**

Error analysis of FEM. Solvability of the Galerkin problem. Finite volumes.
**Monday, March 22**

Boundary conditions in FEM. Smoothing analysis of damped Jacobi. Multigrid idea.
**Wednesday, March 24** - (*Homework 5 due*)

Multigrid algorithms. Elementary convergence analysis.
**Monday, March 29**

*A posteriori* error estimates. Adaptivity. Heat equation. Method of lines.
**Wednesday, March 31**

Lax-Richtmyer theory. Von Neumann stability analysis.
**Monday, April 5** - Easter holiday

**Wednesday, April 7**

Convergence rate for smooth solutions. First order linear hyperbolic systems. Domain of dependence. CFL condition.
**Monday, April 12**

Max-norm stability. Space-time Fourier analysis of constant coefficient linear PDE. Phase and group velocity. Numerical dissipation and dispersion.
**Wednesday, April 14** - (*Homework 6 due*)

Operator splitting. Implicit-explicit methods. Exponential time differencing. Conservation laws. Backward characteristic tracing. Conservative methods. Godunov-type methods. Riemann problem.
**Friday, April 16** - Final exam 2pm–5pm