Concept

Numerical methods for partial differential equations

Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs). In principle, specialized methods for hyperbolic, parabolic or elliptic partial differential equations exist. Finite difference method In this method, functions are represented by their values at certain grid points and derivatives are approximated through differences in these values. Method of lines The method of lines (MOL, NMOL, NUMOL) is a technique for solving partial differential equations (PDEs) in which all dimensions except one are discretized. MOL allows standard, general-purpose methods and software, developed for the numerical integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have been developed over the years in many different programming languages, and some have been published as open source resources. The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary equations can be applied. The method of lines in this context dates back to at least the early 1960s. Finite element method The finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

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https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations

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Source officielle

https://en.wikipedia.org/wiki/Numerical_methods_for_partial_differential_equations

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Concepts associés (5)

Additive Schwarz method

In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down.

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En analyse numérique, la méthode des différences finies est une technique courante de recherche de solutions approchées d'équations aux dérivées partielles qui consiste à résoudre un système de relations (schéma numérique) liant les valeurs des fonctions inconnues en certains points suffisamment proches les uns des autres. Cette méthode apparaît comme étant la plus simple à mettre en œuvre car elle procède en deux étapes : d'une part la discrétisation par différences finies des opérateurs de dérivation/différentiation, d'autre part la convergence du schéma numérique ainsi obtenu lorsque la distance entre les points diminue.

Stabilité numérique

En analyse numérique, une branche des mathématiques, la stabilité numérique est une propriété globale d’un algorithme numérique, une qualité nécessaire pour espérer obtenir des résultats ayant du sens. Une définition rigoureuse de la stabilité dépend du contexte. Elle se réfère à la propagation des erreurs au cours des étapes du calcul, à la capacité de l’algorithme de ne pas trop amplifier d’éventuels écarts, à la précision des résultats obtenus. Le concept de stabilité ne se limite pas aux erreurs d’arrondis et à leurs conséquences.

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Exercices associés (10)

Suspension d une voiture

A car’s suspension is comfortable when it has a vertical oscillation period that our bodies are used to: the oscillation period of walking, Tu2004=u20040.8u2005s.

Find the spring constant k of the suspension in a car of mass $M = 1540\,{\rm kg}$ .
The car lowers when we place a trunk of mass $m = 70\,{\rm kg}$ in it. Find its vertical displacement.

To make the car more comfortable, we add a shock absorber/dampener that creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)

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A sports car is driving around a circular track of radius R with a horizontal velocity of constant norm v. The track is on an incline with angle α from the the horizontal, such that the car is tilted inwards. The static friction coefficient between the tires and the track is μ.

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Question conceptuelle: Coriolis dans votre lavabo

Calculate the order of magnitude of the Coriolis acceleration of the water in your emptying sink. The Coriolis acceleration is expressed with the following equation:

\vec a_{\rm Coriolis}=2\vec{\Omega} \wedge \vec{V} \,

in which Ω⃗ is the Earth’s rotational velocity and V⃗ is the velocity of the water in the sink. We find the following expression for the Coriolis acceleration’s norm:

a_{\rm Coriolis}=2\Omega V \sin\theta \,

with

\Omega=\frac{2\pi }{86400\,\mathrm{s}}

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Notebooks associés (1)

Exploring Chemical Potential for Fermions in 3D at Finite Temperature

This notebook gives a simple illustration to the variation of chemical potential for Fermions in 3D, following the Fermi-Dirac statistics in finite temperature. " In 3D, we have the density of states

\\rho \\propto \\sqrt{E}

. At zero temperature, the distribution of electrons just follows

g(E)=a \\sqrt{E}

and the Fermi energy is determined by the total occupied number

N_{occ}

E_f

is the energy such that

N_{occ}=\\int_{0}^{E_f} g(E) dE

\n", " \n", "In finite temperature, however, o

Personnes associées (32)

Jean-François Molinari

Pablo Antolin Sanchez

Expertise I am currently working on turbulent momentum transport in tokamak using both theoretical and numerical methods.

Olivier Claude Martin Février

Parmenion Mavrikakis

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Exercices associés (10)

Suspension d une voiture

A car’s suspension is comfortable when it has a vertical oscillation period that our bodies are used to: the oscillation period of walking, Tu2004=u20040.8u2005s.

Find the spring constant k of the suspension in a car of mass $M = 1540\,{\rm kg}$ .
The car lowers when we place a trunk of mass $m = 70\,{\rm kg}$ in it. Find its vertical displacement.

To make the car more comfortable, we add a shock absorber/dampener that creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)

Forces de frottement

Draw a diagram, considering the vertical plane perpendicular to the car’s velocity and represent all forces acting on the car.
What’s the condition on v for the car not to drift outwards ?

Numerical application: $g=10\

Question conceptuelle: Coriolis dans votre lavabo

Calculate the order of magnitude of the Coriolis acceleration of the water in your emptying sink. The Coriolis acceleration is expressed with the following equation:

\vec a_{\rm Coriolis}=2\vec{\Omega} \wedge \vec{V} \,

in which Ω⃗ is the Earth’s rotational velocity and V⃗ is the velocity of the water in the sink. We find the following expression for the Coriolis acceleration’s norm:

a_{\rm Coriolis}=2\Omega V \sin\theta \,

with

\Omega=\frac{2\pi }{86400\,\mathrm{s}}

the Earth’s rotational speed,

Afficher plus

Notebooks associés (1)

Exploring Chemical Potential for Fermions in 3D at Finite Temperature

This notebook gives a simple illustration to the variation of chemical potential for Fermions in 3D, following the Fermi-Dirac statistics in finite temperature. " In 3D, we have the density of states

\\rho \\propto \\sqrt{E}

. At zero temperature, the distribution of electrons just follows

g(E)=a \\sqrt{E}

and the Fermi energy is determined by the total occupied number

N_{occ}

E_f

is the energy such that

N_{occ}=\\int_{0}^{E_f} g(E) dE

\n", " \n", "In finite temperature, however, o