Although the Discontinuous Galerkin (dg) method has seen widespread use for compressible flow problems in a single fluid with constant material properties, it has yet to be implemented in a consistent fashion for compressible multiphase flows with shocks and interfaces. Specifically, it is challenging to design a scheme that meets the following requirements: conservation, highorder accuracy in smooth regions and non-oscillatory behavior at discontinuities (in particular, material interfaces). Following the interface-capturing approach of Abgrall [1], we model flows of multiple fluid components or phases using a single equation of state with variable material properties; discontinuities in these properties correspond to interfaces. To represent compressible phenomena in solids, liquids, and gases, we present our analysis for equations of state belonging to the Mie-Grüneisen family. Within the dg framework, we propose a conservative, high-order accurate, and non-oscillatory limiting procedure, verified with simple multifluid and multiphase problems. We show analytically that two key elements are required to prevent spurious pressure oscillations at interfaces and maintain conservation: (i) the transport equation(s) describing the material properties must be solved in a non-conservative weak form, and (ii) the suitable variables must be limited (density, momentum, pressure, and appropriate properties entering the equation of state), coupled with a consistent reconstruction of the energy. Further, we introduce a physics-based discontinuity sensor to apply limiting in a solution-adaptive fashion. We verify this approach with one-and two-dimensional problems with shocks and interfaces, including high pressure and density ratios, for fluids obeying different equations of state to illustrate the robustness and versatility of the method. The algorithm is implemented on parallel graphics processing units (gpu) to achieve high speedup.

Discontinuous Galerkin (DG) methods have recently received much attention because of their portability to complex geometries, scalability in parallel architectures and relatively simple extension to high order. However, their implementation for compressible turbulence problems is not straightforward, e.g., due to parameter-free limiting for orders greater than first and the lack of a consistent high-order diffusion scheme for DG. To address this last point, Van Leer proposed the idea of recovery-based discontinuous Galerkin (RDG) approaches. In the present work, an explicit recovery-based diffusion scheme is developed in three dimensions on a Cartesian grid to solve incompressible and slightly compressible turbulence problems. It is shown that the selection of an optimal recovery and enhancement basis preserves the super-convergence property of this scheme (eighth order for P = 2, where P is the order of the polynomial basis) for the three-dimensional Navier-Stokes equations. Test problems confirm the efficient and accurate application of the present recovery-based discontinuous Galerkin method to incompressible and slightly compressible turbulence problems.

In the present study, a discontinuous Galerkin (DG) framework is developed to simulate chemically reacting flows. The algorithm combines a double-flux method to account for variable thermodynamic properties, a Strang-splitting scheme for the stiff reaction chemistry, a robust WENO-based shock limiter, and the non-linear viscous-diffusive transport is discretized using the BR2 method. The algorithm is verified and validated by considering a series of one-and two-dimensional test cases, and results are compared with self-similarity solutions and experiments to examine critical algorithmic components. These cases include low-Mach deflagration systems and supersonic inviscid and viscous problems. Multi-dimensional configurations consider the shock-flame interaction and detonation initiation process. It is shown that the reactive DG-method provides an accurate description of key-physical mechanisms that control the ignition onset in confined detonation systems.