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

We construct the 2nd-order and 3rd-order cell-centered Lagrangian schemes for 1D elastic-plastic problems with the hypo-elastic constitutive model and the von Mises yield criterion. The basic procedure of the construction is the following: first, we carefully analyze the wave structure of the Riemann problem for elastic-plastic materials and develop a two-rarefaction Riemann solver with elastic waves (TRRSE). Then, based on the developed TRRSE, we propose the 2nd-order and 3rd-order cell-centered Lagrangian schemes for 1D elastic-plastic solid problems. Moreover, we show that our scheme is positivity-preserving, provided the time step is suitably small. A number of numerical experiments are carried out, and the numerical results show that our 3rd-order scheme achieves the desired order of accuracy. Finally, we apply our 2nd-order and 3rd-order schemes to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves, and the numerical results are compared with the reference solution and with the results obtained by other authors. The comparison shows that the current high-order scheme appears to be convergent, stable and essentially non-oscillatory. Moreover, for shock waves the numerical results of our 2nd-order scheme agree very well with those computed by the 2nd-order scheme developed by Maire et al. (2013), while for rarefaction waves the current second-order scheme performs better than Maire et al.'s (2013) second-order scheme. Besides, our third-order scheme performs better than the 2nd-order scheme developed by Maire et al. (2013).