Publication

Convergence Analysis For Spectral Approximation To A Scalar Transport Equation With A Random Wave Speed

2012
Article
Résumé

This paper is concerned with the initial-boundary value problems of scalar transport equations with uncertain transport velocities. It was demonstrated in our earlier works that regularity of the exact solutions in the random spaces (or the parametric spaces) can be determined by the given data. In turn, these regularity results are crucial to convergence analysis for high order numerical methods. In this work, we will prove the spectral convergence of the stochastic Galerkin and collocation methods under some regularity results or assumptions. As our primary goal is to investigate the errors introduced by discretizations in the random space, the errors for solving the corresponding deterministic problems will be neglected.

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Concepts associés (33)
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Les méthodes de Runge-Kutta sont des méthodes d'analyse numérique d'approximation de solutions d'équations différentielles. Elles ont été nommées ainsi en l'honneur des mathématiciens Carl Runge et Martin Wilhelm Kutta, lesquels élaborèrent la méthode en 1901. Ces méthodes reposent sur le principe de l'itération, c'est-à-dire qu'une première estimation de la solution est utilisée pour calculer une seconde estimation, plus précise, et ainsi de suite. Considérons le problème suivant : que l'on va chercher à résoudre en un ensemble discret t < t < .
Vague
Une vague () est une déformation de la surface d'une masse d'eau le plus souvent sous l'effet du vent. À l'interface des deux fluides principaux de la Terre, le vent crée des vagues sur les océans, mers et lacs. Ces mouvements irréguliers se dispersent à la surface de l'eau et sont collectivement appelés état de la mer. D'autres phénomènes, moins fréquents, sont aussi la source de vagues. Ainsi, les séismes majeurs, éruptions volcaniques ou chutes de météorites créent également des vagues appelées tsunamis ou raz-de-marée.
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.
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