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