Sur quelques équations aux dérivées partielles et leur analyse numérique

摘要: In this thesis, four Partial Differential Equations of different nature are studied, numerically or/and theoretically. The first part deals with non-conservative hyperbolic systems in one space dimension. the case systems, several definitions shock waves exist literature, paper, we propose and study a new, very simple genuinely non-linear fields. second is concerned Harmonic Map flow. We build solutions to harmonic map flow from unit disk into sphere which have constant degree, co-rotational symmetric frame. First prove existence such solutions, using time semi-discrete scheme then compute these by moving-mesh method allows us deal singularities. third initial-and-boundary value problem for Kadomtse-Petviashvili II equation posed on strip Dirichlet left boundary condition two kinds conditions right boundary. Moreover treat half plane show result convergence. last part, investigate numerical means conjecture proposed Guy David about new Global Minimizer Mumford-Shah Functional R^3. led spectral Laplace operator Neumann dimensional subdomain S^2 reentrant corners. particular, eigenvector accurate approximations singular coefficients at each corner. For that use Singular Complement Method.