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Lecture

Wave Equation Stability

In course

PHYS-203: Computational physics I

Aborder, formuler et résoudre des problèmes de physique en utilisant des méthodes numériques simples. Comprendre les avantages et les limites de ces méthodes (stabilité, convergence). Illustrer différents sujets de physique traités dans d'autres cours. ...

Description

This lecture covers the stability analysis of the finite difference scheme for the wave equation, exploring the concept of clean modes and resonant excitation. It delves into the explicit diagram with 3 levels, eigenmodes, eigenfrequencies, and the von Neumann stability analysis. The instructor demonstrates the stability condition and the possibility of exponential amplitude growth over time.

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Instructor

Laurent Villard

Originaire de Vauffelin (BE), Laurent Villard est né en 1961. Après avoir obtenu le diplôme d'ingénieur physicien de l'EPFL en 1983, il est assistant au CRPP où il entame une thèse. Ses études portent sur l'étude théorique du chauffage des plasmas par ondes radiofréquences, développant pour cela un code de calcul numérique sous contrat avec le Joint European Torus (JET, Angleterre). Il obtient en 1987 le doctorat ès sciences. En 1988, il est engagé en tant que chargé de cours à l'Université d'Addis Abeba, enseignant l'électromagnétisme et l'électrodynamique. En 1989 et 1990, il rejoint le Polytechnic Education Development Centre de l'Institute of Technology of Bandung (ITB, Indonésie). Ses activités contribuent à la formation et au soutien du personnel enseignant des dix-sept écoles polytechniques nouvellement créées en Indonésie. A la fin 1990, il revient au CRPP en tant que collaborateur scientifique. En 1993 il est nommé professeur assistant. En 1999, il est nommé maître d'enseignement et de recherche, puis, en 2005, professeur titulaire. En collaboration avec JET et General Atomics (USA) il étudie les instabilités qui peuvent être provoquées par les particules énergétiques issues du processus de fusion nucléaire. Avec des scientifiques du Keldysh Institute (Moscou), il s'intéresse au calcul de l'équilibre et de la stabilité de configurations du type tokamak. Enfin, en collaboration notamment avec le Max-Planck Institute fuer Plasma Physik, ses recherches portent sur la simulation numérique, par calcul à haute performance (HPC), de la turbulence dans les plasmas magnétisés.

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Related concepts (32)

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Finite difference method

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

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.

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