IAM562 - Introduction to Scientific Computing II
Instructor(s): Hamdullah Yücel / Ömür Uğur
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.
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
- 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
- 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.
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