High-order in-cell discontinuous reconstruction path-conservative methods for conservative systems in primitive variables

We present a path-conservative methodology for the numerical approximation of conservative systems formulated exclusively in primitive variables, the most physically relevant set of variables. The approach is designed to preserve the correct weak solution while avoiding the spurious oscillations near contact discontinuities that may arise in conservative formulations. Nonconservative products are treated through the selection of a suitable family of paths combined with an in-cell discontinuous reconstruction strategy, which is locally activated near shocks. To choose the path we introduce a compatibility condition between the Rankine-Hugoniot conditions of the non-conservative form, with the exact ones, assumed to be known. In smooth regions and across contact discontinuities, a standard path-conservative scheme is employed. The method is further extended to achieve high-order accuracy. Numerical results for the Euler multimaterial and shear interface model demonstrates correct shock-capturing property, suppression of spurious oscillations due to contact discontinuities, and convergence without using hybrid formulations.