In-cell Discontinuous Reconstruction path-conservative methods for non conservative hyperbolic systems: 2°&high-order extension

Abstract: In the case of systems of conservation laws, in order to ensure the convergence to the right weak solutions of the approximations provided by a method, besides consistency and stability, entropy has to be well controlled. However, in the case of nonconservative systems this is not enough: numerical dissipation effects have to be well controlled. The theoretical framework of path-conservative methods introduced in (Parés,2006) facilitates the design of schemes that are formally consistent with the definition of weak solution based on the well-known theory of Dal Maso, LeFloch and Murat. Recently, in (Chalons, 2019), an in-cell discontinuous reconstruction technique has been added to first-order path-conservative methods that allows one to capture correctly weak solutions with isolated shock waves. In (Pimentel-García et al., 2022) this technique was extended to second-order accuracy and it was added the well-balanced property. The main objectives of this presentation are to do a review on this topic, to talk about its extension to high-order accuracy and to be able to capture well more than one shock wave. This extension is based on the combination of the MOOD strategy and the use of smoothness indicators. Several numerical tests are proposed to validate the methods.