## Courses

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## IAM562 - Introduction to Scientific Computing II

**Credit: **
3(3-0); **ECTS: **
8.0

**Instructor(s): **
Hamdullah Yücel

**Prerequisites: **
Consent of Instructor(s)

#### Course Catalogue Description

Ordinary Differential Equations: Euler’s method, multistep methods, Runge-Kutta methods, stiff equations, adaptivity; Boundary Value Problems: shooting, collocation, Galerkin; Partial Differential Equations: parabolic, elliptic, and hyperbolic equations; Iterative Methods for Sparse Linear Systems: splitting methods, descent methods, conjugate gradients, preconditioners, multigrid methods.

#### Course Objectives

This is a course on scientific computing for ordinary differential equations (ODEs) and partial differential equations (PDEs). It includes the construction, analysis and application of numerical methods for ODEs/PDEs. Objects of this course are:

- to motivate the need for efficient numerical methods for solving differential equations
- to understand basic finite difference methods for partial differential equations
- to analyze consistency, stability, and convergence of the finite difference methods
- to solve system of linear equations numerically using direct and iterative methods
- to implement numerical methods on the computer to solve partial differential equations arising from the sciences and engineering.

#### Course Learning Outcomes

Upon successful completion of this course, the student will be able to:

- understand mathematics-numeric interaction, and how to match numerical method to mathematical properties
- make a good choice of methods for a particular ODE problem
- construct appropriate finite-difference approximations to PDEs
- analyze consistency, stability, and accuracy of a finite difference method
- write programs to solve ODEs/PDEs by finite difference methods
- solve challenging problems that are either purely mathematical or practical from various disciplines.

#### Tentative (Weekly) Outline

- Introduction to ODEs and Euler’s Method
- Multistep Methods for ODEs
- Runge-Kutta Methods for ODEs
- Stiff Equations and Adaptivity in Time
- Boundary Value Problems: shooting, collocation, Galerkin
- Introduction to PDEs
- Parabolic Equations
- Parabolic Equations: methods of lines
- Elliptic Equations: iterative solvers
- Hyperbolic Equations
- Iterative Solvers: splitting methods, descent methods, conjugate gradients
- Iterative Solvers: preconditioners, GMRES algorithm
- Iterative Solvers: multigrid methods
- Review of the Topic Material

#### Course Textbook(s)

- A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2009.
- R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady- State and Time-Dependent Problems, SIAM, 2007.

#### Supplementary Materials and Resources

- Books:
- M. T. Heat, Scientific Computing, McGraw Hill, 1997.
- A. Quarterioni, R. Sacco, and F. Salari, Numerical Mathematics, Springer, 2000.
- A. Quarterioni and F. Salari, Scientific Computing with MATLAB and Octave, Springer-Verlag, 2006.
- David F. Griffiths, John W. Dold, David J. Silvester, Essential Partial Differential Equations: Analytical and Computational Aspects, Springer, 2015.
- Resources:
- MATLAB
- htps://odtuclass.metu.edu.tr

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