A unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier-Stokes equations for mixed grids. The CPR framework can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a differential formulation without explicit volume or surface integrations. Several test cases are computed to demonstrate its performance. Abstract The compact high-order 'Spectral

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... Volume Method' (SVM, Wang (2002) ) designed for conservation laws on unstructured grids is presented. Its spectral reconstruction is exposed briefly and its applications to the Euler equations are presented through several test cases to assess its accuracy and stability. Comparisons with classical methods such as MUSCL show the superiority of SVM. The SVM method arises as a high-order accurate scheme, geometrically flexible and computationally efficient. Abstract The aim of this contribution is to develop a high order numerical scheme for simulating compressible multiphase flows. For reaching high order, we propose to use the Runge-Kutta Discontinuous Galerkin method. The development of such a method is not straightforward, because it was originally developed for conservative systems, whereas the system of interest is not conservative. We show how to circumvent this difficulty, and prove the accuracy and the robustness of our method on one and two dimensional numerical tests.