A high-order cell-centered Lagrangian scheme for one-dimensional elastic–plastic problems

•Develop a two-rarefaction Riemann solver (TRRSE) for 1D elastic–plastic flows.•Propose high-order cell-centered Lagrangian schemes for 1D elastic–plastic problems.•Show how to limit the time step to keep our high-order scheme positivity-preserving.•Our 3rd-order scheme seems to be convergent, stable and essentially non-oscillatory.•TRRSE performs better than the node Riemann solvers in resolving rarefaction waves.