Analyse numérique des schémas quadratisés. Application à la simulation de la corde de piano non-linéaire
Guillaume Castera, Juliette Chabassier · HAL (Le Centre pour la Communication Scientifique Directe) · 2023
In this report, we present energy quadratization techniques for a Hamiltonian system of nonlinear wave equations formulated at order 2 in time. On a generic system, we present the so-called Invariant Energy Quadratization (IEQ) and Scalar Auxiliary Variable (SAV) methods as well as their energy conservation properties and discretization strategies in space and time. Unlike the iterative techniques commonly used for nonlinear systems to guarantee certain invariances, these two methods lead to algorithms whose complexity is known in advance and rely on the simple inversion of a linear system at each time step. In spite of an unconditional stability and an attractive complexity, the literature mentions problematic application cases with an uncontrolled accuracy.The numerical properties (stability, consistency and uniform convergence in time with respect to the CFL) of schemes obtained by hybridization between theta scheme and quadratization are studied for two classes of nonlinear terms: a nonlinearity concerning the solution field, and a nonlinearity concerning its gradient.These results are then applied to a geometrically exact nonlinear piano string for which numerical results are presented. The influence of the discretization parameters is studied and related to the theoretical results. The choices for the best accuracy or computational cost are illustrated. Some parameters can induce the space-time non-convergence of the schemes for a nonlinearity on the gradient, as it is the case for the piano string.