Numerical Methods II, Spring 2018

Course Details and Syllabus

Assignments, workload, and testing

There will be written and programming assignments. These will be weekly at first but tapering toward the end to allow time for project work. Late assignments may be accepted at the discression of the instructor, but there will be a lateness penalty. Each homework assignment is designed to take 10 hours or less. Please let the instructor know if your are spending significantly more time than this. There will be a group project with a presentation during finals week and written report due then. It should take about half the semester to complete the project. There will be no quizzes or written exam.

Source materials

See the Resources page.


Most class communication will be on the NYU Classes site through the class message board there. Check the message board before starting any homework assignment, as there may be corrections or hints. Please post questions or comments about the assignments or the class there. Grades will be posted there. You may also communicate with fellow students, setting up group meetings or exchanging ideas about homework. Please email the instructor only for personal matters (schedule an appointment, request to submit an assignment late, etc.).


The final grade will be determined by the grades on the assignments and the project, each counting for about half the total. Written assignment solutions should be submitted in hard copy in class. Some coding assignments will ask students to upload their code and output to the NYU Classes course site.

Collaboration and cheating policy

Students are encouraged to discuss homework exercises with each other. Each student must write the solutions himself or herself. Copying of solutions or allowing others to copy your solutions is considered cheating and will be handled according to NYU cheating policies and the more stringent policies of the Mathematics and Computer Science Departments. Code sharing is not allowed. Unless explicitly told otherwise, you must create every character of code you use. All code for the group project must be created by group members, unless the instructor gives specific permission otherwise.

Outline (tentative)

Segment Topics
1 Time stepping for ODE, Runge Kutta methods. Order of accuracy, Taylor series analysis, higher order methods. Convergence theory, computational and theoretical error analysis. What can be computed for chaotic systems.
2 More on Runge Kutta methods. Stiff problems and implicit methods for them. Stability regions for common Runge Kutta methods. Adaptive methods. The Verlet method for Newtonian dynamics.
3 Linear multistep methods. Adams Bashforth/Moulton and Nyström methods. Stability for linear recurrence relations. Stability regions for common linear multistep methods. Statements (not proofs) of some Dahlquist barrier theorems.
4 Hyperbolic PDE. Quick review of gas dynamics (compressible Euler equations) and acoustics (wave equation) in one dimension. Characteristic modes and speeds. Finite difference marching methods. Lax Wendroff method (Euler equations) and leapfrog (wave equation). Boundary conditions.
5 Stability and convergence. Review of discrete Fourier analysis, the von Neumann stability condition. The geometric CFL condition. Examples of unstable and stable schemes. Semi-discrete schemes and ODE stability regions. Upwind schemes.
6 More on time dependent problems. Diffusion, explicit time step constraints, implicit methods. Local conservation laws, conservation form, numerical flux functions.
7 Laplace, steady advection/diffusion and other elliptic problems. Finite difference and conservative discretization. Consistency/stability arguments and stability norms. Jacobi and Gauss Seidel iteration schemes.
8 Finite element discretizations of elliptic PDE. Variational and weak formulations. Sobolev space $H_1$. Finite dimensional subspaces from triangulations. Interpolation approximations and shape regularity.
9 Iterative methods for linear symmetric problems. Conjugate gradient orthogonalization and the role of condition number. Preconditioners.
10 Multigrid methods for elliptic problems.