## Math 228A - Numerical Solution of Differential Equations (Fall 2016)

#### Course details

• Description: Theory and practical methods for numerical solution of differential equations. Ordinary differential equations: Runge-Kutta and multistep methods, stability theory, stiff equations, boundary value problems. Partial differential equations: Finite difference and spectral methods for elliptic, parabolic and hyperbolic equations, stability, accuracy and convergence, von Neumann analysis and CFL conditions.

• Prerequisites: Math 128A or equivalent knowledge of undergraduate numerical analysis, MATLAB or equivalent programming experience.

• Lectures: MWF 10am - 11am, room 247 Cory.

Office hours: Mon 12:30pm - 2pm and Fri 11am - 12:30pm in 1089 Evans.

• GSI: Noble Macfarlane, ntmacfarlane@berkeley.edu, 1044 Evans.
Office hours: Thu 9am - 11am in 1044 Evans.

#### Other resources

• Grades will be based entirely on the 7 problem sets. Please start early, it might be hard to get help the last few days before the due dates. Problem sets can be handwritten or typed, but must be clear and well organized. They can be handed in during the lecture on the due date or submitted online through bCourses. Computer codes should be submitted online, according to the instructions in each problem set.

It is encouraged to discuss the problem sets with other current students, but each student must write his/her own solutions and computer codes, and understand all the details of them. It is not permitted to consult any solutions from courses given in previous years (including both reference solutions and other students' problem sets).

#### Schedule

1 W 8/24 Overview, ODEs (IVPs, BVPs), PDEs
2 F 8/26 IVP Theory L:5.1-5.2, Str:1-2, I:1.1,A.2.3
3 M 8/29 Basic Numerical Methods for IVPs L:5.3 , I:1.2,2.1,3.2
4 W 8/31 MATLAB, the ODE Suite ML, Mol, Sha
5 F 9/02 Convergence of Euler's Method Str:3.2, I:1.2
M 9/05 Labor day - No lecture
6 W 9/07 Stiff Equations L:8.1-8.2, Str:3.3, I:4.1-4.2
7 F 9/09 Linear Stability Theory, A-stability L:7.1-7.6, Str:3.5-3.6, I:4.2 PS1 Due
8 M 9/12 Implicit Methods Str:3.4, I:7.1-7.3
9 W 9/14 Taylor Series Methods L:5.6, Str:2.6
10 F 9/16 Explicit Runge-Kutta (ERK) Methods L:5.7, Str:4.1-4.4, I:3.2
11 M 9/19 Implicit Runge-Kutta (IRK) Methods I:3.3
12 W 9/21 Runge-Kutta Order Conditions Str:4.5, I:3.C
13 F 9/23 Gaussian Quadrature I:3.1 PS2 Due
14 M 9/26 IRK Methods from Collocation I:3.4
15 W 9/28 Lecture canceled
16 F 9/30 A-stability of Runge-Kutta Methods Str:4.6-4.8, I:4.3
17 M 10/03 L-stability and B-stability L:8.3, Str:4.9-4.10
18 W 10/05 Linear Multisteps Methods, Adams Methods L:5.9, Str:10.1-10.3, I:2.1
19 F 10/07 Backward Differentiation Formulae Methods L:8.4, Str:10.4, I:2.3 PS3 Due
20 M 10/10 Order and Convergence of Multistep Methods L:6.4, Str:11.1-11.5, I:2.2
21 W 10/12 A-stability of Multistep Methods L:7.3, Str:11.6, I:4.4
22 F 10/14 Stepsize and Error Control I:6.1-6.2
23 M 10/17 Embedded Runge-Kutta Methods Str:5.1-5.4, I:6.3
24 W 10/19 Finite Difference Approximations L:1.1-1.5, I:8.1
25 F 10/21 Convergence of BVPs L:2.1-2.10 PS4 Due
26 M 10/24 Boundary Layers and Nonuniform Grids L:2.12-2.17
27 W 10/26 FD methods for elliptic problems I L:3
28 F 10/28 FD methods for elliptic problems II L:3
29 M 10/31 FD methods for elliptic problems III L:3
30 W 11/02 FD methods for parabolic problems I L:9
31 F 11/04 FD methods for parabolic problems II L:9 [PS5 Due]
32 M 11/07 FD methods for parabolic problems III L:9
33 W 11/09 FD methods for hyperbolic problems I L:10
F 11/11 Veterans day - No lecture
34 M 11/14 FD methods for hyperbolic problems II L:10
35 W 11/16 FD methods for hyperbolic problems III L:10
36 F 11/18 Compact finite difference schemes I Lele92 [PS6 Due]
37 M 11/21 Compact finite difference schemes II Lele92
W 11/23 Thanksgiving - No lecture
F 11/25 Thanksgiving - No lecture
38 M 11/28 Spectral methods for BVPs I Tre
39 W 11/30 Spectral methods for BVPs II Tre
40 F 12/02 Spectral methods for BVPs III Tre [PS7 Due]
RRR week 12/5-12/9 - No lectures