A hybrid reconstructed discontinuous Galerkin and continuous Galerkin method based on an incremental pressure projection formulation, termed rDG(P n P m)+CG(P n) in this paper, is developed for solving the unsteady incompressible Navier-Stokes equations on unstructured grids. In this method, a reconstructed discontinuous Galerkin method (rDG(P n P m)) is used to discretize the velocity and a standard continuous Galerkin method (CG(P n)) is used to approximate the pressure. The rDG(P n P m)+CG(P n) method is designed to increase the accuracy of the hybrid DG(P n)+CG(P n) method and yet still satisfy Ladyženskaja-Babuška-Brezzi (LBB) condition, thus avoiding the pressure checkerboard instability. An upwind method is used to discretize the nonlinear convective fluxes in the momentum equations in order to suppress spurious oscillations in the velocity field. A number of incompressible flow problems for a variety of flow conditions are computed to numerically assess the spatial order of convergence of the rDG(
Discontinuous Galerkin (DG) methods have been well established for single material hydrodynamics. However, consistent DG discretizations for non-equilibrium multi-material (more than two materials) hydrodynamics have not been extensively studied. In this work, a novel reconstructed discontinuous Galerkin (rDG) method for the single-velocity multi-material system is presented. The multi-material system being considered assumes stiff velocity relaxation, but does not assume pressure and temperature equilibrium between the multiple materials. A second-order DG(P 1 ) method and a third-order least-squares based rDG(P 1 P 2 ) are used to discretize this system in space, and a third-order TVD Runge-Kutta method is used to integrate in time. A well-balanced DG discretization of the non-conservative system is presented, and is verified by numerical test problems. Further, a consistent interface treatment is implemented, which ensures strict conservation of material masses and total energy. Numerical tests indicate that the DG and rDG methods are indeed second-and third-order accurate. Comparisons with the second-order finite volume method show that the DG and rDG methods are able to capture the interfaces more sharply. The DG and rDG methods are also more accurate in the single-material regions of the flow. This work focuses on the general multidimensional rDG formulation of the non-equilibrium multi-material system and a study of properties of the method via one-dimensional numerical experiments. The results from this research will be the foundation for a multi-dimensional high-order rDG method for multi-material hydrodynamics.
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