This is the primary course website for Advanced Numerical Analysis (MATH 269A), Fall 2024.

There will be several assignments involving both theoretical and computational exercises. These will be collected on Friday in class.

- Homework 1, Due 10/11, in class.

The following is a tentative plan.

- 1 Business. Course overview - why 269A?
- 2 Euler's method. Alternate derivations of Euler's method.
- 3 Generalizations of Euler's method. ODE classification and ODE standard form.
- 4 Local truncation error, Euler's method error bound.
- 5 Euler's method error bound. Order notation. Huen's method. Huen's method error bound.
- 6 Error bound for Huen's method. ODE methods for systems of ODEs.
- 7 Higher order methods. Runge-Kutta methods.
- 8 Asymptotic error estimates and estimating rates of convergence.
- 9 Estimating rates of convergence - practical details.
- 10 Timestep selection for qualitatively correct approximate solutions.
- 11 Timestep selection for "model problem". Intervals of absolute stability.
- 12 Timestep selection for "model problem" and for general problems.
- 13 Examples of timestep estimation for general problems.
- 14 Timestep estimation for linear constant coefficient systems.
- 15 Timestep estimation for general systems.
- 16 Timestep estimation for general systems.
- 17 Identifying stiff ODE's.
- 18 Methods for stiff differential equations. Implicit methods.
- 19 Implicit equations solution techniques.
- 20 Implicit equations solution techniques.
- 21 Identifying stiffness, consequences of inexact implicit solves.
- 22 Choice of initial iterate. General concepts for adaptive timestep determination.
- 23 Adaptive timestepping. Estimation of local truncation error.
- 24 Estimation of local truncation error using stepsize doubling.
- 25 Linear multi-step methods (LMM), derivation, estimation of local truncation error.
- 26 Linear multi-step methods (LMM). Linear constant coefficient difference equations. Convergence results.
- 27 LMM convergence results. LMM absolute stability.
- 28 End of quarter summary.

Ernest K. Ryu, Mathematical Sciences 7619B,

Office hours: Wednesday 12:00–1:30pm

Zheng Tan

Office hours: Tuesday and Thursday 3:00–4:00pm

Monday, Wednesday, and Friday 4:00–4:50pm at Mathematical Sciences 5118.

Tuesday 4:00–4:50pm at Mathematical Sciences 5118. Run by the TA

This class will have in-person closed-book hand-written (no computers) midterm and final exams.

- Midterm exam: Date and location TBD.
- Final exam: 12/11, 8:00–11:00am, location TBD

Homework 30%, midterm exam 30%, final exam 40%.

Systems of ordinary and/or partial differential equations form the basis for nearly all mathematical models used in the physical, social and engineering sciences. Most of the equations that arise cannot be solved "by hand" and require the use of numerical methods to obtain solutions. The focus of the Math 269 series of courses is on the numerical methods used to create approximate solutions of systems of ordinary and partial differential equations. In 269A, the principle goals consist of identifying, analyzing, and implementing the most commonly used numerical techniques to construct approximate solutions of systems of ordinary differential equations (ODE's). A variety of single step and multistep methods will be covered as well as general theoretical aspects that, when understood, help with the tasks of method selection and implementation. Methods for "stiff systems" and adaptive timestepping methods will be covered in the latter part of the quarter.

For graduate students, it is recommended that the students have taken at least one undergraduate course in numerical analysis, preferably two (e.g. Math 151AB), as well as an upper-division undergraduate course in linear algebra. The assignments will involve programming so an undergraduate course in programming is recommended.

The basic qualification for undergraduates to enroll in 269A, if there is space, is that you have completed 151AB and 131AB, and are maintaining a good GP by showing an unofficial transcript to me.

Ascher and Petzold, *Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations*, SIAM, 1998.