A Godunov-type method in Lagrangian coordinates for computing linearly-perturbed planar-symmetric flows of gas dynamics

Linear stability studies of complex flows require that efficient numerical methods be devised for predicting growth rates of multi-dimensional perturbations. For one-dimensional (1D) basic flows – i.e. of planar, cylindrical or spherical symmetry – a general numerical approach is viable which consists in solving simultaneously the one-dimensional equations of gas dynamics and their linearized forms for three-dimensional perturbations. Extensions of artificial viscosity methods have thus been used in the past. More recently [Equations aux dérivées partielles et applications, articles dédiés à J.-L. Lions, 1998], Godunov-type schemes for single-fluid flows of gas dynamics and magnetohydrodynamics have been proposed. Pursuing this effort, we introduce, within the Lagrangian perturbation approach, a class of Godunov-type schemes which is well suited for solving multi-material problems of gas dynamics. These schemes are developed here for the planar-symmetric case and comprise two second-order extensions. The numerical capabilities of these methods are illustrated by computations of Richtmyer–Meshkov instabilities occurring at a single material interface. A systematic comparison of numerically computed growth rates with results of the linear theory for the Richtmyer–Meshkov instability is provided.