A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces

• Published in: Journal of computational physics 2015-01, Vol.280 (C), p.489-509
• Main Authors: Henry de Frahan, Marc T., Varadan, Sreenivas, Johnsen, Eric
• Format: Article
• Language: English
• Subjects: Algorithms; Compressibility; Conservation; Constraining; Density; Discontinuous Galerkin; Equations of state; Galerkin methods; High-order limiting; Interface capturing; Interface oscillations; Mie–Grüneisen equation; Multiphase and multicomponent flows; Multiphase flow; Non-conservative equations
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2014.09.030

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, high-order 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.