The subject of the course is quite broad for one semester since the area of Scientific Computing is developing very fast as it has a lot of applications in many sciences. Numerical simulation is a significant tool for the study of scientific problems which arise in sciences like Physics, Chemistry, Geology, Biology, Economics etc. Most of these problems end up with the solution of a system of Ordinary or Partial Differential equations which can only be solved numerically. The goal is the student to obtain the necessary knowledge in order to be able to develop not only the most efficient numerical algorithm but also its corresponding software for the study of scientific problems by simulation.

The course covers the study of numerical solution methods for Ordinary and Partial Differential equations which consist the core of the Scientific Computations. Special emphasis is given on the practical application. In particular the syllabus consists of the following:

**Part I** Numerical methods for the solution of Ordinary Differential Equations : methods of Euler, Taylor, Truncation and roundoff errors, Consistency, Convergence and Stability. Second, third and fourth order Runge – Kutta methods, Errors and Stability. Multiple step methods: Adams-Bashforth, Predictor-Corrector, Consistency, Convergence and Stiffness. Boundary Value Problems : Method of finite differences.

**Part II** Introduction to finite differences. Numerical solution of Parabolic Equations : One dimensional Parabolic equations, Direct methods, Crank-Nicolson, Convergence, Stability. Two dimensional Parabolic equations : Direct methods, Iterative methods (ADI). Three dimensional Parabolic equations. Numerical solution of Elliptic equations : Iterative methods (SOR, SSOR). Numerical solution of Hyperbolic equations : Direct and Iterative methods.

- M. Vraxatis, Numerical Analysis : Ordinary Differential Equations, Kleidarithmos, 2012.https://service.eudoxus.gr/search/#s/%CE%92%CF%81%CE%B1%CF%87%CE%B1%CF%84%CE%B7%CF%82/0
- Nikolaos Missirlis, Numerical Analysis : An algorithmic approach, Publication by National and Kapodistrian University of Athens, 2017.https://service.eudoxus.gr/search/#s/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CE%B7%CF%84%CE%B9%CE%BA%CE%AE%20%CE%91%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7/0