Semester | Mand. | Option | Filières |
MA1, MA3 | x | ||
MA1, MA3 | x | A |
Lecture | 2 weekly |
Exercises | 2 weekly |
Study plan | Semester | Mand. | Option | Filières |
Computational science and Engineering 2018-19 | MA1 | x | A | |
Computational science and Engineering 2018-19 | MA3 | x | A | |
Ingénierie Mathématiques 2018-19 | MA1 | x | A | |
Ingénierie Mathématiques 2018-19 | MA3 | x | A | |
Mathématiques HEP 2018-19 | MA1 | x | ||
Mathématiques HEP 2018-19 | MA3 | x | ||
Mathématiques Master 2018-19 | MA1 | x | A | |
Mathématiques Master 2018-19 | MA3 | x | A |
Cursus | Sem. | Type |
---|---|---|
Computational science and Engineering | MA1, MA3 | Opt. |
Ing.-math | MA1, MA3 | Opt. |
Mathematics for teaching | MA1, MA3 | Opt. |
Mathématicien | MA1, MA3 | Opt. |
The modeling of many problems in the applied sciences and engineering is based on concepts of conservation of mass, momentum and energy, leading to systems of conservation laws. Prominent examples are the Maxwell equations of electromagnetics, the Euler and Navier-Stokes equations of fluid dynamics and equations of elasticity and the systems of magnetohydrodynamics of plasma physics.
In this course we develop, analyse and apply computational methods suitable for solving systems of conservation laws. We begin by discussing fundamental properties of conservation laws, including their ability to generate non-smooth solutions - shocks - from smooth initial conditions, leading to the introduction of weak solutions and entropy conditions.
We initiate the discussion of computational methods by finite difference methods for conservation laws and introduce the concept of monotone schemes and the consequences of this. This is followed by a thorough discussion of finite volume methods, including Godunov's methods, monotone fluxes, accuracy, and stability of such methods.
We then begin the discussion of methods of higher order accuracy, first by understanding why higher order accuracy is of interest. This sets the stage for the introduction of limiter based schemes with a particular emphasis on the MUSCL scheme. Subsequently, we discuss in detail essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) methods and their fundamental properties. Higher order in time is achieved through the development of strongly stable Runge-Kutta methods (SSP-RK).
As a final extension to more geneal problems, we develop discontinuous Galerkin methods as a very general and robust high-order accurate extension of finite volume methods. We study these methods in some detail, including their mathematical properties.
Throughout the course there will be an emphasis on mastering mathematical as well as computational aspects of the different methods.
Conservation laws, finite volume methods, MUSCL scheme, ENO/WENO methods, discontinuous Galerkin methods
A course in partial differential equations and their numerical approximation.
Linear partial differential equations, numerical approximation, stability, convergence. Basic methods for solving ordinary differential equations and computational linear algebra.
The class will be given as a lecture class with in-class computational experiments to support the analysis.
Development of computational methods for conservation laws, their analysis, implementation and use for solving application examples of increasing complexity.
There will be 3 required small reports to be handed in during the class. These will be examined as part of the final oral examination and will count for 50% of the overall grade.
Dans le cas de l’art. 3 al. 5 du Règlement de section, l’enseignant décide de la forme de l’examen qu’il communique aux étudiants concernés.
The class will use the text
J.S. Hesthaven, Numerical Methods for Conservation Laws: From Analysis to Algorithms. SIAM Publishing, 2017.
optionnel
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Cours
Lecture
Heure(s) hebdo x 14 semaines
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Exercices
Exercises
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Hour(s) per week x 14 weeks
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Oral
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HIV
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Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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Oral
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optionnel
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Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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HIV
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optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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Oral
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Session hiver
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HIV
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optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
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HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
HIV
optionnel
optionnel
Cours
Lecture
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Exercices
Exercises
Heure(s) hebdo x 14 semaines
Hour(s) per week x 14 weeks
H hebdo
H hebdo
Oral
Oral
Session hiver
Session winter
HIV
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\x26#949;
JS_fin
CODE169976220
BOOLEAN_Symbol
BaliseJS_debut
\x5blig\x5d
BaliseJS_fin
Balise_debut
[lig]
Balise_fin
FO_debut
[inline]=›  
FO_fin
HTML_debut
›  
HTML_fin
JS_debut
\x26#x203A;\x26#160;\x26#160;
JS_fin
CODE148292899
BOOLEAN_Symbol
BaliseJS_debut
\x5bEPSILON\x5d
BaliseJS_fin
Balise_debut
[EPSILON]
Balise_fin
FO_debut
[inline]=Ε
FO_fin
HTML_debut
Ε
HTML_fin
JS_debut
\x26#917;
JS_fin
BALISETXT_SAUT
BOOLEAN_Symbol
BaliseJS_debut
\x5bbr\x2f\x5d
BaliseJS_fin
Balise_debut
[br/]
Balise_fin
FO_debut
FO_fin
HTML_debut
<br/>
HTML_fin
JS_debut
\x3cbr\x2f\x3e
JS_fin