3. Objectives: In numerical computation, one always has a choice of methods that can be used to solve a given problem. It often happens that, out of a two methods that look nearly identical at the first glance, one may fail while the other one will produce good results. The key objectives of the course is to learn how to avoid the mistakes in choosing the right method for a specific problem and how to design and analyze numerical methods suitable for solving partial differential equations with the help of finite differences.

4. Structure: Two class meetings per week, homework every two to three weeks, final exam. All classroom lectures will be videotaped. Recordings can subsequently be viewed on Panopto.

5. Grading Policy: 70% of the credit will be earned during the semester, while the remaining 30% will come from the final exam, which has been scheduled for Tuesday, December the 12th, between 12:00pm and 2:30pm (see general examination schedule for Fall 2023). A number of problem solving, computer, and reading homework assignments will be handed out during the semester toward the aforementioned 70% of the total credit. All homework papers must be computer typeset. Numerical results shall be presented clearly in the form of tables and/or plots. Homework submitted after the due date may still be accepted but will earn a lower grade unless there is a documented excused absence. In the latter case, assignments must still be completed but may be submitted at a later date with no penalty.

6. Classroom attendance does not contribute to the overall grade directly, but is recommended on the substance. Viewing the videotaped lectures may help.


7. Topics to be covered (ideally):

Chapter 9: Introductory material on finite-difference methods for ordinary differential equations. The notions of convergence, consistency, and stability. Convergence as an implication of consistency and stability. Specific examples, Runge-Kutta methods. Methods for solving boundary-value problems. Saturation of finite-difference methods by smoothness. A brief account of spectral methods.

Chapter 10: Finite-difference methods for partial differential equations. The relation between consistency, stability, and convergence. The Courant, Friedrichs, and Lewy condition. Maximum principles. Several different approaches to constructing the schemes. Stability analysis for various settings. Stability with respect to the initial data -- the von Neumann spectral criterion. Various norms. Energy methods. Systems vs. scalar equations. Variable coefficients, dissipation, Kreiss' condition. Initial boundary value problems; Babenko-Gelfand and Godunov-Ryaben'kii stability criteria; the GKS theory. Implicit and explicit schemes for the heat equation.

Chapter 11: Schemes for computing discontinuous solutions of hyperbolic conservation laws.

Chapter 12 and Sections 5.7 & 6.4: Schemes for elliptic equations. Maximum principle. Discrete Fourier series. Multigrid methods. A brief notion of finite elements.