Monday, January 4IVP for ODE. The Picard-Lindelöf theorem. Lipschitz condition. Stability. Euler's method.
Wednesday, January 6One-step methods. Global error analysis. Backward Euler.
Monday, January 11RK family. Order conditions. Adaptive methods. Embedded RK pairs.
Wednesday, January 13LMM. Consistency. Order conditions. Zero-stability. Root condition. Convergence.
Monday, January 18Dahlquist 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 25Stability region of LMM. Root locus curve. Dahlquist second barrier. A(α)-stability. L-stability.
Wednesday, January 27Stability function of RK methods. Collocation.
Monday, February 1Gauss, 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 8Gradient flows. Conservative systems. Hamiltonian systems. N-body problem. Energy and momentum. Angular momentum. Partitioned RK. Reversibility.
Wednesday, February 10Adjoint. Symmetric methods. Composition. Reversible methods. Symplectic flows.
Monday, February 15Two 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 1Stability in 2-norm. Conditioning of the discretized system.
Wednesday, March 3 - (Homework 4 due)Collocation and Petrov-Galerkin frameworks. Fourier series.
Monday, March 8Convergence of Fourier series.
Wednesday, March 10Fourier-Galerkin methods. Trapezoidal quadrature and DFT. FFT. Chebyshev polynomials.
Monday, March 15Multidimensional BVP. Weak formulation. Finite elements. Element-by-element assembly.
Wednesday, March 17Error analysis of FEM. Solvability of the Galerkin problem. Finite volumes.
Monday, March 22Boundary conditions in FEM. Smoothing analysis of damped Jacobi. Multigrid idea.
Wednesday, March 24 - (Homework 5 due)Multigrid algorithms. Elementary convergence analysis.
Monday, March 29A posteriori error estimates. Adaptivity. Heat equation. Method of lines.
Wednesday, March 31Lax-Richtmyer theory. Von Neumann stability analysis.
Monday, April 5 - Easter holiday
Wednesday, April 7Convergence rate for smooth solutions. First order linear hyperbolic systems. Domain of dependence. CFL condition.
Monday, April 12Max-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