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Identical coursesNumerical Simulation of Differential Equations (B-KUL-H0M80A)
Cannot be taken as part of an examination contract
Samaey Giovanni (coordinator) | Feppon Florian Aims
Due to their complexity, the differential equations that engineers and scientists are confronted with usually do not allow for an exact analytical solution. One is then obliged to compute approximate numeral solutions. Via some characteristic model problems, the students in this course learn how to transform a differential equation into a discrete numerical problem that can be solved on a computer. After this course, the student will be able to:
- describe standard discretisation techniques for ordinary differential equations (linear multistep methods, Runge-Kutta methods) and partial differential equations (finite differences,finite elements and finite volumes)
- analyse the convergence properties of these methods (consistency, stability, convergence, accuracy) and variants
- explain how different properties of the method affect computational cost (implicit vs. explicit methods, solution of nonlinear systems)
- discuss the suitability of these methods for specific types of problems (stiff or geometric ordinary differential equations; parabolic, hyperbolic and elliptic partial differential equations)
- implement these methods for a concrete application, and compare and explain their behaviour in terms of the properties of the method and the problem under study.
Previous knowledge
The student should have a basic knowledge of calculus, including differential equations, and numerical mathematics.
Identical courses
This course is identical to the following courses:
H03D7A : Numerieke simulatie van differentiaalvergelijkingen (No longer offered this academic year)
Is included in these courses of study
- Master in de ingenieurswetenschappen: wiskundige ingenieurstechnieken (Leuven) 120 ects.
- Master in de wiskunde (Leuven) 120 ects.
- Master of Mathematics (Leuven) 120 ects.
- Courses for Exchange Students Faculty of Engineering Science (Leuven)
- Master of Mathematical Engineering (Leuven) 120 ects.
- Master in de fysica (Leuven) (Optie fysica in de maatschappij) 120 ects.
- Master of Actuarial and Financial Engineering (Leuven) 120 ects.
Activities
4.5 ects. Numerical Simulation of Differential Equations: Lecture (B-KUL-H0M80a)
Content
Part I: Ordinary differential equations
- Forward and backward Euler method, trapezoidal rul
- Order of a method / consistency / convergence
- Stiffness, stability
- Geometric integration
- Higher-order methods: linear multistep methods and Runge-Kuttamethods
- Splitting methods
Part II: Elliptic partial differential equations
- Finite differences: order and convergence
- Finite elements
- Spectral methods
Part III: Parabolic partial differential equations
- Finite differences for the one-dimensional heat equation
- Finite differences for higher-dimensional parabolic problems
- Finite elements and spectral methods for parabolic problems
Part IV: Hyperbolic partial differential equations
- Finite difference for the linear advection equation
- Non-linear hyperbolic conservation laws and finite volume methods
Course material
Study cost: 1-10 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)
Own course material, distributed via Toledo.
Format: more information
Lectures, exercise sessions and practical assignments are integrated in 20 contact moments of 2h.
These contact moments are prepared by the students via short implementation assignments and numerical experiments. These assignments are the starting point for the instruction of new material.
1.5 ects. Numerical Simulation of Differential Equations: Exercise Sessions and Projects (B-KUL-H0M81a)
Content
Lectures, exercise sessions and practical assignments are integrated in 20 contact moments of 2h.
These contact moments are prepared by the students via short implementation assignments and numerical experiments. These assignments are the starting point for the instruction of new material.
Course material
Handbook/articles and literature/Toledo.
Evaluation
Evaluation: Numerical Simulation of Differential Equations (B-KUL-H2M80a)
Explanation
For more information on question types and grading, see Toledo/
Information about retaking exams
If the student failed the practicals, he/she will get a new assignment.