An efficient implicit scheme for the multimaterial Euler equations in Lagrangian coordinates

• Efficient fully implicit discretization of the multimaterial Euler equations written inLagrangian (mass) coordinates This work presents an fully implicit discretization strategy based on a non-conservative waveequation-based predictor, bypasses the stringent timestep restrictions typical of explicit methods, particularly for stiff materials and high-density ratio systems, while requiring only the solution of simple tridiagonal linear systems for a predictor pressure. • Filtering for density and pressure oscillations at material interfaces and contacts Themethod introduces filtering techniques to suppress spurious oscillations in density and pressure at material interfaces and contacts. • Mass-preserving, interface-preserving smoothly varied Lagrangian grid generation strategies A simple generation technique for monodimensional Lagrangian grids in mass coordinates is proposed, imposing mass conservation and preserving the position of material interfaces, while allowing smooth variation of grid spacing, which is important when dealing with large density ratios. • Applications to multilayer media with stiff materials and large density ratios The method is applied to stratified systems with large density and stiffness differences, and tested with CFL numbers up to 109. • Robustness with respect to large shocks Extensive testing shows that the method can handle large shocks while maintaining stability, despite being conceived mainly to treat less violent phenomena. Stratified fluids composed of a sequence of alternate layers show interesting macroscopic properties, which may be quite different from those of the individual constituent fluids. On a macroscopic scale, such systems can be considered a sort of fluid metamaterial. In many cases each fluid layer can be described by Euler equations following the stiffened gas equation of state. The computation of detailed numerical solutions of such stratified material poses several challenges, first and foremost the issue of artificial smearing of material parameters across interface boundaries. Lagrangian schemes completely eliminate this issue, but at the cost of rather stringent time step restrictions. In this work we introduce an implicit numerical method for the multimaterial Euler equations in Lagrangian coordinates. The implicit discretization is aimed at bypassing the prohibitive time step restrictions present in flows with stratified media, where one of the materials is particularly dense, or rigid (or both). This is the case for flows of water-air mixtures, air-granular media, or similar high density ratio systems. We will present the novel discretisation approach, which makes extensive use of the remarkable structure of the governing equations in Lagrangian coordinates to find the solution by means of a single implicit discrete wave equation for the pressure field, yielding a symmetric positive definite structure and thus a particularly efficient algorithm. Additionally, we will introduce simple filtering strategies for counteracting the emergence of pressure or density oscillations typically encountered in multimaterial flows, and will present results concerning the robustness, accuracy, and performance of the proposed method, including applications to stratified media with high density and stiffness ratios.