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

Abstract

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. Here, 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), coupledmore » 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. Lastly, the algorithm is implemented on parallel graphics processing units (gpu) to achieve high speedup.« less

Authors:
ORCiD logo [1];  [1];  [1]
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  1. University of Michigan, Ann Arbor, MI (United States)
Publication Date:
Research Org.:
Univ. of Michigan, Ann Arbor, MI (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); US Department of the Navy, Office of Naval Research (ONR); National Science Foundation (NSF)
OSTI Identifier:
1418534
Alternate Identifier(s):
OSTI ID: 1247001
Grant/Contract Number:  
FC52-08NA28616; N00014-12-1-0751; CBET 1253157; DEFC52-08NA28616
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 280; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

Henry de Frahan, Marc T., Varadan, Sreenivas, and Johnsen, Eric. A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces. United States: N. p., 2014. Web. doi:10.1016/j.jcp.2014.09.030.
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Henry de Frahan, Marc T., Varadan, Sreenivas, & Johnsen, Eric. A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces. United States. https://doi.org/10.1016/j.jcp.2014.09.030
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Henry de Frahan, Marc T., Varadan, Sreenivas, and Johnsen, Eric. Thu . "A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces". United States. https://doi.org/10.1016/j.jcp.2014.09.030. https://www.osti.gov/servlets/purl/1418534.
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@article{osti_1418534,
title = {A new limiting procedure for discontinuous Galerkin methods applied to compressible multiphase flows with shocks and interfaces},
author = {Henry de Frahan, Marc T. and Varadan, Sreenivas and Johnsen, Eric},
abstractNote = {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. Here, 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. Lastly, the algorithm is implemented on parallel graphics processing units (gpu) to achieve high speedup.},
doi = {10.1016/j.jcp.2014.09.030},
journal = {Journal of Computational Physics},
number = C,
volume = 280,
place = {United States},
year = {Thu Oct 02 00:00:00 EDT 2014},
month = {Thu Oct 02 00:00:00 EDT 2014}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1016/j.jcp.2014.09.030
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Cited by: 31 works
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Figures / Tables:

Figure 1 Figure 1: Pressure profile at t = 2 for the advection of a sharp material interface (P = 2, Δ$\mathcal{x}$ = 1/128). Solid red: conservative transport equation and limiting of the conserved variables (fully conservative approach). Dashed green: non-conservative equation and limiting of the conserved variables. Dash-dotted blue: non-conservative equationmore » and modified limiting (our approach).« less
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