Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Hu, Xiangyu; Adams, Nikolaus A.; Shu, Chi‐Wang

doi:10.1016/j.jcp.2013.01.024

Cited by 193 publications

(145 citation statements)

References 24 publications

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“…Based on certain Gauss-Lobatto quadratures and positivity preserving flux, in order to preserve the positivity of density and pressure fields in numerical calculations, a limiting procedure under a CFL-like condition [17,36] presented by Zhang and Shu [13][14][15] and simplified by Wang et al [16] is developed in limiting the solution with positivity-preserving flux and the TVB limiter.…”

Section: Positivity-preserving Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Liu

Qiu

Goman

et al. 2016

Numer. Math. Theory Methods Appl.

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Abstract. In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

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Section: Positivity-preserving Methodsmentioning

confidence: 99%

“…The second example is a high Mach number astrophysical jet problem, which was used as an example in [13,15,36].…”

Section: High Mach Number Astrophysical Jetmentioning

confidence: 99%

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Liu

Qiu

Goman

et al. 2016

Numer. Math. Theory Methods Appl.

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“…There exist in the literature two types of positivitypreserving flux limiters, which can be borrowed and extended to the GRHD case, including the cut-off flux limiter [15] and the parametrized flux limiter [6,17,18,40,41]. The extension of the cut-off limiter to the GRHD case is similar to the special RHD case [35].…”

Section: Pcp Flux Limitermentioning

confidence: 99%

Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

2017

Phys. Rev. D

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The paper develops high-order physical-constraint-preserving (PCP) methods for general relativistic hydrodynamic (GRHD) equations, equipped with a general equation of state. Here the physical constraints, describing the admissible states of GRHD, are referred to the subluminal constraint on the fluid velocity and the positivity of the density, pressure and specific internal energy. Preserving these constraints is very important for robust computations, otherwise violating one of them will lead to the ill-posed problem and numerical instability. To overcome the difficulties arising from the inherent strong nonlinearity contained in the constraints, we derive an equivalent definition of the admissible states. Using this definition, we prove the convexity, scaling invariance and LaxFriedrichs (LxF) splitting property of the admissible state set G, and discover the dependence of G on the spacetime metric. Unfortunately, such dependence yields the non-equivalence of G at different points in curved spacetime, and invalidates the convexity of G in analyzing PCP schemes. This obstacle is effectively overcame by introducing a new formulation of the GRHD equations. Based on this formulation and the above theories, a first-order LxF scheme is designed on general unstructured mesh and rigorously proved to be PCP under a CFL condition. With two types of PCP limiting procedures, we design high-order, provably (not probably) PCP methods under discretization on the proposed new formulation. These high-order methods include the PCP finite difference, finite volume and discontinuous Galerkin methods.

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“…For instance, at this year's SIAM CSE conference 2 , there was a dedicated session on Positivity Preserving and Invariant Domain Preserving Methods, where many of the talks addressed these issues for high-order methods. Some of the more recent publications include [151,152,61,19,121]. Hence, simpler methods may be more convenient when we are dealing with very complex flows, such as multiphase flow models or flows with strong source terms.…”

Section: Motivationmentioning

confidence: 99%

Large Time Step Roe scheme for a common 1D two-fluid model

Prebeg

Flåtten

Müller

2017

Applied Mathematical Modelling

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Large Time Step (LTS) finite volume methods are presented, analyzed and applied to one-dimensional hyperbolic conservation laws.The HLL (Harten-Lax-van Leer) and HLLC (HLL-Contact) schemes are extended to LTS-HLL(C) schemes. The LTS-HLL-type schemes for scalar conservation laws are defined in the numerical viscosity, flux-difference splitting and wave propagation form, and TVD conditions on wave velocity estimates are determined. It is shown that the LTS-HLL-type schemes contain some already existing LTS methods such as the LTS-Roe and LTS-Lax-Friedrichs, and that they allow us to deduce LTS extensions of other methods, such as the Rusanov and Engquist-Osher schemes. The LTS-HLL(C) schemes for systems of conservation laws are developed by standard field-by-field decomposition. Numerical results suggest that the computational efficiency of LTS methods depends on the type of problem under consideration. In certain cases LTS methods for nonlinear systems of equations yield increased accuracy, efficiency and convergence rate compared to standard methods.Entropy stability of LTS-HLL-type schemes is analyzed. We use modified equation analysis to gain insight into numerical diffusion in LTS-HLLtype schemes and to conjecture about entropy stability of LTS methods. Theoretical results obtained through the modified equation analysis are in agreement with numerical results for both scalar conservation laws and the Euler equations.Positivity preservation in LTS methods is investigated. We show that the positivity preserving conditions for LTS methods are stronger than for standard methods. We describe different ways positivity can be lost in LTS-HLL-type schemes for the Euler equations, and we propose a simple way to increase the robustness of the LTS-HLL-type schemes by adding numerical diffusion. Numerical investigations show that LTS-HLL-type schemes successfully handle test cases for positivity preservation, and where the positivity is lost we can improve the robustness by adding numerical diffusion.In addition, we applied the LTS-Roe scheme to a two-fluid model and focused on the treatment of the boundary conditions and the source terms. It is shown that stability and accuracy of the solution can be greatly improved by appropriate treatment of boundary conditions and source terms.i

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Positivity-preserving method for high-order conservative schemes solving compressible Euler equations

Cited by 193 publications

References 24 publications

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics

Large Time Step Roe scheme for a common 1D two-fluid model

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