WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

Xiong, Tao; Shu, Chi‐Wang; Zhang, Mengping

doi:10.1007/s10915-012-9578-7

Cited by 14 publications

(10 citation statements)

References 57 publications

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“…Multicomponent flows are thus modelled as a medium described by a single equation of state with variable material properties that must be transported in an appropriate form; discontinuities in these properties correspond to interfaces. Saurel and Abgrall [14] and Abgrall and Karni [15] detailed various approaches to solve this problem in the fv context, which can be extended to high-order accuracy [16,17] and finite differences [18,19]. Such methods also conserve the total mass, momentum, and energy in the system.…”

Section: Introductionmentioning

confidence: 99%

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

Frahan

Varadan

Johnsen

2015

Journal of Computational Physics

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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.

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Section: Introductionmentioning

confidence: 99%

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

Frahan

Varadan

Johnsen

2015

Journal of Computational Physics

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“…In [10], we have developed a WENO scheme with subcell resolution for computing nonconservative Euler equations with applications to one-dimensional compressible two-medium flows. This is a first step towards the development of a robust and accurate solver for multi-medium flows.…”

Section: Summary Of Researchmentioning

confidence: 99%

High Order Accurate Weighted Essentially Non-oscillatory Algorithms for Shock Calculations

Shu¹

2012

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“…Therefore, it is worthwhile to seek better numerical solutions for such a set of flow equations. In fact, the nonconservative formulations exist in a broader context and have demonstrated usefulness in a number of applications …”

Section: Introductionmentioning

confidence: 99%

Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

2019

Numerical Methods in Fluids

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Summary In this paper, discontinuous Galerkin (DG) discretization schemes for Navier‐Stokes equations of a nonconservative form was proposed, in which the constitutional equation for energy conservation is written in terms of pressure. The primary focus is to address the treatment of nonconservative products in the pressure equation, for which we formulate four different scheme variants by using the path‐conservative scheme or solely regarding the nonconservative products as source terms. In addition to the complete description of the discretization formulations, we highlight the positivity‐preserving properties of the proposed nonconservative DG schemes. The performance of the four schemes are thoroughly assessed and tested with the consideration of a series of classical flow configurations that involve steady/unsteady, inviscid/viscous, and laminar/turbulence flow physics. It is found that two out of the four schemes are able to provide the optimal convergence and similar accuracies as the fully conservative formulation. The nonconservative formulations provide an alternative way to model fluid flows with substantial thermodynamic and compositional complexities. The present work aims to lay the theoretical foundation for further algorithmic extensions to simulations of such fluid flows of practical relevance.

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WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

Cited by 14 publications

References 57 publications

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

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

High Order Accurate Weighted Essentially Non-oscillatory Algorithms for Shock Calculations

Development of a nonconservative discontinuous Galerkin formulation for simulations of unsteady and turbulent flows

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