Syllabus

Please note that these schedules are tentative and they may be modified if needed.

Week 1: Motivation and introduction to differential equations and concepts of accuracy and stability. ODEs: 1-step Euler methods: derivation; explicit; implicit; accuracy and stability; trapezoidal rule. Methods of derivation: Taylor series; undetermined coefficients; polynomial fitting.

Week 2: Motivation and introduction to differential equations and concepts of accuracy and stability. ODEs: 1-step Euler methods: derivation; explicit; implicit; accuracy and stability; trapezoidal rule. Methods of derivation: Taylor series; undetermined coefficients; polynomial fitting.

Week 3: ODEs: Compound methods: Predictor-Corrector; Runge-Kutta. Extrapolation methods: Richardson; Gragg; Bulirsch-Stoer. Systems of equations: stiffness.

Week 4: ODEs: Two-point boundary-value problems: shooting; relaxation; projection.

Week 5: PDEs: Classification. General concepts of finite-difference approximations. Methods for parabolic equations: Forward-Time Centered-Space (FTCS; explicit); Backward-Time Centered-Space (BTCS: implicit); Crank-Nicolson. Von Neumann stability analysis.

Week 6: PDEs: Methods for hyperbolic equations: Failure of FTCS; Lax method. Courant-Friedrich-Lewy (CFL) condition. Other types of errors: phase (dispersion errors); nonlinear (numerical diffusion); transport errors (leading to upwind schemes).

Week7: PDEs: Methods for hyperbolic equations (continued): Second order methods: Leapfrog; 2 step Lax-Wendroff. Higher order finite-difference methods: 3rd order Essentially Non-Oscillatory; 5th order Weighted ENO (WENO).

Week 8: PDEs: Methods for elliptic equations: 5-point finite-difference stencil. Review of direct solvers: Gaussian elimination; LU. Review of iterative solvers: Jacobi iteration; Gauss-Seidel iteration; Successive Over-Relaxation; Conjugate Gradient; preconditioning; accelerators (red-black coloring; multigrid). Dirichlet/Neumann boundary conditions.

Week 9: Multidimensional problems: Alternating Direction Implicit (ADI). Other rapid elliptic solvers: Spectral methods, including Fourier; Chebyshev; pseudo-spectral methods; FFTs; dealiasing; difference between Galerkin, tau, collocation methods.

Week 10: A glimpse at other methods (at discretion of instructor): Finite element, finite volume, etc.