A graduate-level course on numerical methods for partial differential equations. Learners are expected to have taken a first undergraduate course in numerical methods such as HMTHCS212 and a first course in PDEs such as HMTH407. Topics include finite difference and spectral methods for time-dependent and stationary partial differential equations. Error and stability analysis, Lax Equivalence Theorem. Implementation of finite difference and spectral methods in MATLAB.
Learners are expected to study the assigned resources prior to the class discussions. Reading material will be posted online on this site, and on WhatsApp group chat.
| Lesson | Date | Topics | Comments | Readings | Exercises |
|---|---|---|---|---|---|
| 1 | October 10 | Review of PDEs, fundamental concepts and examples, well-posed IVPs |
Evans' notes on PDEs Ascher: pg. 1-12 Sections 1.1 (pg. 1) - 1.1.4 (pg. 12) |
Review: Ascher pg. 32: 0 (a), (b), (c), (d), (e) 1, 3, 4. |
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| 2 | October 12 | finite differences, stability ideas, review of mathematical foundations |
Ascher: pg. 12-31 Sections 1.2(pg. 12) - 1.3.6 (pg. 32) |
Review: Ascher pg. 32: 0 (f), (g), (h), (i) 5, 6 |
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| 3 | October 14 | finite difference for the advection equation, amplification factors, Von Neumann analysis, CFL condition |
Ascher: pg. 12-31 Sections 1.2(pg. 12) - 1.3.6 (pg. 32) |
Review: Ascher pg. 32: 0 (f), (g), (h), (i) 5, 6 |
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| 4 | October 17 | finite difference for parabolic PDEs heat equation in 1D, truncation errors, convergence |
Python codes for the heat equation Explicit difference scheme Fully Implict difference scheme |
Morton and Mayers: pg. 7-36 Sections 2.1(pg. 7) - 2.7 (pg. 33) |
Review: Morton and Mayers pg. 56: 2.3, 2.6, 2.7, 2.9 |
| 5 | October 19 | parabolic PDEs in 1D, the \(\theta\)-method, Crank-Nicolson |
Assignment 1 due 29 October
Solutions |
Morton and Mayers: pg. 36-52 Sections 2.8(pg. 36) - 2.15 (pg. 52) |
Review: Morton and Mayers pg. 56: 2.3, 2.6, 2.7, 2.9 |
| 6 | October 21 | parabolic PDEs in 2 and 3D, ADI Method in 2D, Douglas-Rachford, Peaceman-Rachford schemes |
truncation error of the ADI scheme stability of the ADI scheme |
Morton and Mayers: pg. 62-71 Sections 3.1(pg. 62) - 3.3 (pg. 71) |
Review: Morton and Mayers pg. 83: 3.1, 3.2, 3.3 |
| 7 | October 24 | consistency, convergence, stability Lax Equivalence Theorem |
Morton and Mayers: pg. 151 - 166 Sections 5.1(pg. 151) - 5.6 (pg. 160) |
Review: Morton and Mayers pg. 190: 5.1, 5.2, 5.4 |
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| 8 | October 27 | consistency, convergence, stability calculating stability conditions |
Morton and Mayers: pg. 151-166 Sections 5.1(pg. 151) - 5.6 (pg. 166 ) |
Review: Morton and Mayers pg. 190: 5.1, 5.2, 5.4 |
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| 9 | October 31 | hyperbolic PDEs, dissipation adding dissipation to schemes |
Python codes for the advection equation Upwind Leapfrog Lax Wendroff Lax Friedrichs Crank-Nicolson |
Strikwerda: pg. 121-125 Sections 5.1(pg. 121-125) |
Review: Strikwerda pg. 125: 5.1.1, 5.1.2, 5.1.4, 5.1.5 |
| 10 | Nov 2 | hyperbolic PDEs, dispersion group velocity, wave packets |
Assignment 2: due 18 Nov
Solutions |
Strikwerda: pg. 125-134 Sections 5.2-5.3(pg. 125-134) Ascher: pg 211-217 Section 7.1 (pg. 211-217) |
Review: Strikwerda pg. 129: 5.3.1, 5.3.2, 5.3.3, 5.3.4 Review: Ascher: pg 246 0(a), (b), (c), (d), 1, 2, 3(a), (b). |
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Final Exam |