CLASS SCHEDULE MTH 4/552 Winter 2025 See Syllabus and RESOURCES == week 1 1. 1/6: Introduction. Linear and nonlinear IVP/BVP. [TEXTBOOK Chapter 5] 2. 1/8: Lipschitz condition and implication for E/U of solutions to IVP. Examples how to find Lipschitz condition. Stability and continuous dependence. Gronwall's lemma: differential form. 3. 1/10: Applying Gronwall's lemma to prove uniqueness. First numerical schemes: Forward Euler (FE) and Backward Euler (BE). Sources of error: one-step (LTE) and accumulation. == week 2 4. 1/13: Meet and greet your fellow students. Coding loops (needed for piecewise functions). How to evaluate the bounds (number M) and Lipschitz condition (number L) on some region D around initial condition. Examples. 5. 1/15: Interpretation of FE scheme as using left rectangular rule of integration, or as replacing derivative by a finite difference quotient. Taylor expansions for functions in C^k, with Lagrange remainder. How to evaluate the LTE (one step error). Examples of second order schemes. 1/16 HW1 due 6. 1/17: One-step schemes viewed through the lenses of numerical integration of the right hand side function: left, right, midpoint, trapezoidal rules: their accuracy depends on which polynomials they integrate exactly. Explicit vs implicit schemes: revisiting the concept with examples. Introducing the Heun method. Worksheet on the steps to establish LTE ( to be continued). == week 3 No class on 1/20 (MLK Holiday) 7. 1/22 Complete the worksheet on LTE for the Heun method. Examples how to design schemes/approximations by matching expansions and identifying the coefficients. 8. 1/24 Richardson extrapolation: another way to design schemes when finding higher order approximations to u'(t). How to assemble information about the errors and produce error plots that show consistency (convergence) order. See http://sites.science.oregonstate.edu/~mpesz/latex/mth-homework-template-Peszynska.pdf for a template example. EXTRA office hours: (MATLAB) Programming "clinic" 4:00-6:00; Kidd 033. Bring questions and work in progress on HW. == week 4 9. 1/27 Yet another way of constructing higher order schemes by approximating the integral of f(t,u(t)) in multiple stages. Runge-Kutta schemes with Butcher tableaus. HW2 due. 10. 1/29 Order of consistency of RK schemes defined with Butcher tableaus: conditions on the coefficients. Examples of unscrambling the schemes, and checking the order conditions. 11. 1/31 More on RK: implicit vs explicit. Order (diagonally implicit and explicit) and order barriers. Examples. What does it take to complete a time step when using an implicit scheme such as BE: (fixed point-simple-Picard) iteration. Computational cost and round-off error. == week 5 12. 2/3 How to estimate the "one step" error (or the LTE). Difference between the so-called a-priori (assumes known analytical solution) and a-posteriori (found from numerical solution only) error estimates. Embedded RK schemes reuse calculated stage approximations to obtain a higher order scheme solution which can be used to estimate the error. Connection to Richardson extrapolation methods. 13. 2/5 How to construct Adams family of schemes: calculate the integral of the polynomial approximations of the right hand side. Due: HW3 14. 2/7: REVIEW. See REVIEW-sheet.pdf EXTRA OFFICE HOURS Friday 2/7 15:00-15:50. == week 6 15. 2/10 MIDTERM in class. 16. 2/12 Midterm discussion. Calculation of total error for the FE scheme: how the LTE error accumulates from time step to time step. Worksheet MTH_452_552_worksheet-onFE-error.pdf 17. 2/14 Zero-stability of single-step schemes. Definition of LMM. [TEXTBOOK, section 5.9]. Root condition for characteristic polynomials. (Extra credit/optional) HW-worksheet due (on paper) in class. HW-worksheet (optional), due on paper in class on Friday 2/14 == week 7 18. 2/17 Continue zero-stability. Solving difference equations. Dahlquist theorem. [TEXTBOOK, Chapter 6]. 19. 2/19 Continue convergence for schemes other than FE. Try error calculations for schemes other than FE: observe how the error accumulates. Develop Lipschitz condition for the right hand side function \Psi for FE, Heun. Define growth factors R(z) for z=h\lambda. How to use these to study if the qualitative nature of the numerical solution agrees with that for the true solution. HW4 due 20. 2/21 Determining and plotting stability regions for single-step schemes in R and in C. Reminder how to work with complex numbers. Absolute stability region definition for single step schemes. == week 8 21. 2/24 Studying R^ABS for various schemes. Plotting functions on R^2 and on C. Plotting R^ABS. [TEXTBOOK, Chapter 7]. 22. 2/26 A-stable and L-stable schemes [TEXTBOOK, Sections 8.1-8.4] . Examples how to find h so that z = h\lambda is in R^ABS. 23. 2/28 Examples of systems; how to implement a scheme in MATLAB for a linear system with constant coefficients. == week 9 24. 3/3 Wrap-up numerical schemes for IVP: how to choose a method? How to chose a time step? Due: HW5 Start BVP problems (second order, linear, with constant coefficients). Boundary conditions of Dirichlet, Neumann, Robin type. Examples when a solution exists, does not exist, and exists but is not unique. 25. 3/5 "BVP are hard" (as opposed to IVP). Worksheet and a discussion how to approximate u''(x) with a finite difference quotient: C^4 smoothness required!. (You can refresh your memory of finite difference quotients by reading [TEXTBOOK, Chapter 1]. ) 26. 3/7 How to set up the numerical solution of a BVP discretized by FD in the form of a linear system of equations (worksheet) . Properties of the system for Dirichlet b.c. and for Neumann b.c. Symmetric positive definite (spd) matrices. [TEXTBOOK, Chapter 2: focus on sections 2.1-2.10. Read 2.11-2.15 lightly] == week 10 27. 3/10 Error for a finite difference scheme for a BVP: (i) deriving error equation, (ii) error estimate (discussion of vector and matrix norms to follow; see [TEXTBOOK, Appendix A]). Implementation of a scheme for second order BVP. Contrast between the analysis for IVP and BVP. 28. 3/12. Finish the error estimate for simple BVP. Due: HW6 29. 3/14 Review. Review worksheet. == week 11 (FINALS) EXTRA OFFICE HOURS Monday 3/17 12:00-14:00. Final exam Wed 03/19.