On well-balanced implicit-explicit Lagrange-projection schemes for two-layer shallow water equations

•Two-layer shallow water equations in Lagrangian coordinates.•Implicit-explicit well-balanced Lagrange-projection schemes.•Well-balanced approximate Riemann solver based on a relaxation approach.•Numerical simulations in subsonic regimes for two-layer flows. This works concerns the study of well-balanced Lagrange-projection schemes applied to the two-layer shallow water system. In particular, a formulation of the mathematical model in Lagrangian coordinates is proposed. The HLL method is then applied to a simplified version of the resulting Lagrangian system. Furthermore, based on the acoustic-transport splitting interpretation, another approximate Riemann solver for the acoustic-Lagrangian step is described. Both an explicit and an implicit-explicit method are proposed, where the latter can allow very fast simulations in sub-critical regimes. Finally, we show numerical simulations in which the outputs are compared with the IFCP method’s results.