## Numerical Methods for PDEs 2011

Lecturer |
Prof. Petros Koumoutsakos Dr. Shilpa Khatri |

Class Times |
Monday, 15.00 - 17.00 in CAB G11 Friday, 13.00 - 15.00 in HG D3.2 |

Assistants |
Dr. Diane de Zelicourt, CAB F81, This email address is being protected from spambots. You need JavaScript enabled to view it.Babak Hejazialhosseini Eivind Fonn, CAB F81, This email address is being protected from spambots. You need JavaScript enabled to view it. , HG J59, This email address is being protected from spambots. You need JavaScript enabled to view it. |

TA Sessions |
Thursday, 13.00 - 15.00 in HG D7.1Thursday, 13.00 - 15.00 in HG D7.2 Friday, 10.00 - 12.00 in HG D3.2 |

Office Hours(in assistant offices) |
Monday, 17.00 - 18.00 Wednesday, 17.00 - 18.00 |

__Note:__Please make sure to include [NPDE] in the subject field of your emails.

__Latest Updates:__- 08/10: HW10 solution code. The latest version uploaded.
- 08/29: 6 solution code. On page 3, "Now, we must subsitutute ...", beta should have been written to be proportional to the time step and not the square of the time step. The manipulations are still correct.

__Course Overview:__

Introduction to numerical methods for the solution of Partial Differential Equations for scientific and engineering problems.

Topics include: Finite element, finite difference, finite volume and particle methods.

Focus on algorithms, efficient computer implementation, scope and limitations of the methods.

__Goal:__

**Introduce students to problem solving in Computational Science and Engineering. The students will acquire skills in :**

**Efficient implementation of numerical methods for the solution of partial differential equations in scientific and engineering applications****Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations**

**Selection and assessment of numerical methods in light of the predictions of theory and the demands of the applications**

**Examples will be drawn from areas such as Physics, Computer Graphics and Electromagnetics.**

__Content (and number of dedicated lectures):__

**I.**

**Introductio**

**n (1 lecture)**

II. Parabolic Differential Equations (8 lectures)

II. Parabolic Differential Equations (8 lectures)

**- Finite Difference Methods**

III. Elliptic Differential Equations (8 lectures)

- Finite Element Method (FEM)

- Variational analysis

- Variational analysis

IV. Hyperbolic equations for Conservation Laws (7 lectures)

- Finite volume methods

V. Summary and Other Methods (1 lecture)

- Particle Methods

**Lecture Schedule and Notes:**

Lecture slides will be made available to the audience as the semester progresses.Lecture slides will be made available to the audience as the semester progresses.

Week | Monday | Friday | ||

1 (21/02) |
Class Cancelled |
00. Introduction: Why study PDEs? (notes) |
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2 (28/02) |
01. PDE of the 1st and 2nd order: General form & Characteristics (notes) |
02. Characteristics. Introduction to parabolic equations. Derivation of the diffusion eq. (notes) - quadratic eq. in geometry (wikipedia) - random walk (applet) |
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3 (07/03) |
03. Parabolic Equations (notes) - analytical considerations - finite differences - semi-discretization |
04. Fourier Series & Transforms (notes) |
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4 (14/03) |
05. DFT - Part A (notes, slides) |
06. DFT - Part B (notes)FFT (notes, slides) |
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5 (21/03) |
07. ODEs (notes) |
08. ODEs, stability and stiff ODEs (notes, slides) |
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6 (28/03) |
09. Parabolic PDEs in 1D - Stability analysis: eigenvalue & Von Neuman (notes) |
10. Parabolic PDEs in 1D: Modified equation, Crank-Nicholson, Dufort-Frankel (notes) |
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7 (04/04) |
11. Parabolic PDEs in 2D or 3D - Alternative Direction Implicit (notes) |
12. Approximate Factorization (notes)Notes on Newton-Raphson (notes) |
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8 (11/04) |
NO CLASS(SechselĂ¤uten) |
13. Elliptic equations (notes)(suppl.) |
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9 (18/04) |
NO CLASS(Easter holidays) |
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10 (25/04) |
NO CLASS(Easter holidays) |
NO CLASS(Easter holidays) |
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11 (02/05) |
MID-TERMSolution: (pdf) (code) |
15. Iterative methods: Point Jacobi, Gauss-Seidel, SOR (notes)(slides) |
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12 (09/05) |
16. FFT and Fast Poisson solvers (notes) |
17. Finite Element Method 1D (notes)(slides) |
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13 (16/05) |
18. Finite Element Method 2D (notes)(slides) |
19. 1D Hyperbolic Equations (notes) |
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14 (23/05) |
20. Finite Volume, general considerations (notes) |
21. Finite Volume Schemes (notes) |
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15 (30/05) |
22. Rieman problem, Godunov scheme (notes)(slides) |

__Homework:__

*Homeworks and solutions will be provided as the semester progresses.*

**HW 1: review of vector and differential calculus. slides, HW01****HW 2****: Characteristics. (****HW02****) (Solution)****HW 3: Integrals and introduction to Fourier. (****HW03****) (****Code03****) (Solution)**

**HW 4: Two weeks project (HW04)****HW 5: Stability of ODEs (analytical and numerical): (HW05) (Code05_simple-integrator)**

**Optional: types.h and hints to implement the time integrator as an operator (Code05_integration-operator)**

**TA notes (pdf)**

**Solution (pdf) (Code)**

**HW 6: Heat equation (analytical and numerical): (HW06) (Code06)****HW 7: Diffusion/Heat equation in 2D:****(HW07)****(Code07)****Solution (pdf) (Code)****Glut libraries (full intsallation package) (Compiled for Optimus)****(Compiled for CAB)**

**HW 8: Point Jacobi, Gauss-Seidel and SOR****(HW08)****(Code08)**

**HW 9: Fourier Series to solve Poisson Equation****(HW09)****(Code09)****HW 10: FEM****(HW10)****(Code10)****HW 11: Finite Volume****(HW11)****(Code11)**

__Grades and Testat:__**Testat: To get a "PASS" you need to have a score of 75% (or more) on 50% of the HW assignements****Grade: The total grade will be computed as a weighted average of your homeworks, mid-term and final exams, with the following weights:****Homeworks: 25%****Mid-term exam: 25%****Final exam: 50%**

**Exam dates****Mid-term exam: May, 2nd 2011**

**Final exam: during semester session (August or September)**

**Books and References used for the Lectures:**DFT and FFT

- Introduction to Applied Mathematics. G Strang
- Advanced Engineering Mathematics. M. Greenberg
- Wikipedia (FFT, Cooley-Tukey Algorithm)

- A First Course in the Numerical Analysis of Differential Equations. A. Isereles
- Numerical Methods for Engineering Application. JH Ferziger 1998
- Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations. Ascher and Petzold
- Numerical solution of initial value problems in differential algebraic equations. Brennan, Campbell and Petzold

- Fundamentals of Engineering Numerical Analysis. P. Moin 2010

- Finite Volume Methods for Hyperbolic Problems (Chap 3). RJ LeVeque 2004

**Additional Reading:**- Solving PDEs in C++: numerical methods in a unified object-oriented approach. Y Shapira
- Python Scripting for Computational Science. H P Langtangen
- S. Childress. Notes on Traffic Flow. 2005 (pdf)