Computational methods of science
Faculteit  Science and Engineering 
Jaar  2017/18 
Vakcode  WICMS08 
Vaknaam  Computational methods of science 
Niveau(s)  bachelor 
Voertaal  Engels 
Periode  semester I a 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Computational methods of science  
Leerdoelen  1. The student is able to describe the methods mentioned in the course description. 2. The student is able to state the keywords and basic notions of numerical methods for PDEs 3. Student is able to apply techniques from (vector) calculus, linear algebra and elementary functional analysis to analyze numerical methods for PDEs. 4. The student is able to argue why analysis of numerical methods is relevant for both the development of new methods and its reliable application to practical problems 5. The students is able to perform simple calculations with a finite element package. 6. The student is able to program simple methods for PDEs in MATLAB. Students can explain the behavior observed in the results . 7. The student is able to decompose a PDE problem in several subproblems, all having their own solution method. 

Omschrijving  Engineering problems are often studied by means of computer simulation. This course concerns numerical techniques for applied problems governed by partial differential equations. Many techniques will be seen at work in the program COMSOL, a finite element package for modelling. During the course predefined problems from electromagnetics, fluid dynamics and mechanics have to be worked out to master the theory. Moreover, the students will be challenged to ask research questions themselves while doing the lab exercises. The following aspects are considered: 1. Classification of first and second order PDEs 2. Discretization of partial differential equations, in general consistency, stability and convergence (Lax equivalence theorem) a. Elliptic equations Discretization of selfadjoint operators, variational problems, discrete maximum principle, finitedifference, finitevolume and finiteelement approximations, properties of discrete operators, treatment of boundary conditions, local and global discretization error, higherorder discretizations b. Parabolic and hyperbolic equations , Von Neumann stability analysis, CFL condition, method of lines, explicit and implicit schemes: Forward and backward Euler, BDF methods, CrankNicolson. 3. Solution of sparse systems a. reordering techniques for direct methods, b. iterative methods: stationary methods (Jacobi, Gauss Seidel, SOR, multigrid method) and instationary methods based on Krylov subspaces (GG, GMRES, BiCG), preconditioning. c. Complexity of solution methods for sparse systems 4. Eigenvalue problems a. Methods to determine a number of eigenvalues: Power method, orthogonal subspace iterative methods, Arnoldi’s Method and Lanczos’ method 

Uren per week  
Onderwijsvorm 
Hoorcollege (LC), Practisch werk (PRC)
(The Lab Sessions are mandatory. Deadlines will be set; not meeting a deadline will result in zero points for the associated exercises) 

Toetsvorm 
Practisch werk (PR), Schriftelijk tentamen (WE)
(For each practical the student can earn 0.6 points. There are 5 practicals which lead to 3.0 points in total. The remaining 6.0 points can be earned by the written exam. Together with the bonus 1.0 this adds up to 10.0 points. The side condition is that the total of the practicals should be at least 1.5 and the points for the written exam at least 3.0.) 

Vaksoort  bachelor  
Coördinator  dr. ir. F.W. Wubs  
Docent(en)  dr. ir. F.W. Wubs  
Verplichte literatuur 


Entreevoorwaarden  Prior knowledge and skill in handling basic numerical techniques and programming in MATLAB as taught in Numerical Mathematics 1 are required. Furthermore, basic knowledge of linear algebra and calculus is necessary.  
Opmerkingen  The course is a necessary introduction for those who want to follow the course Computational Fluid Dynamics or Numerical Bifurcation Analysis of Large Scale Systems. In general for anyone who wants to create numerical algorithms for PDE problems.  
Opgenomen in 
