Finite volume local evolution Galerkin method for two-dimensional relativistic hydrodynamics

2017

Self Cite

This paper presents second-order accurate genuine BGK (Bhatnagar-Gross-Krook) schemes in the framework of finite volume method for the ultra-relativistic flows. Different from the existing kinetic flux-vector splitting (KFVS) or BGK-type schemes for the ultra-relativistic Euler equations, the present genuine BGK schemes are derived from the analytical solution of the Anderson-Witting model, which is given for the first time and includes the "genuine" particle collisions in the gas transport process. The BGK schemes for the ultra-relativistic viscous flows are also developed and two examples of ultra-relativistic viscous flow are designed. Several 1D and 2D numerical experiments are conducted to demonstrate that the proposed BGK schemes not only are accurate and stable in simulating ultra-relativistic inviscid and viscous flows, but also have higher resolution at the contact discontinuity than the KFVS or BGK-type schemes.

Section: Introductionmentioning

confidence: 99%

Second-order accurate genuine BGK schemes for the ultra-relativistic flow simulations

Chen

Kuang

2017

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“…Since then, they become the benchmark tests for verifying the accuracy and resolution of numerical schemes, see [24,16,12,31,29]. In the first configuration, we take Γ = 1.4 and assume that the initial data −0.7259 of fluid velocity in (4.6) are perturbed to −0.7259 + 0.1ξ.…”

Section: D Casementioning

confidence: 99%

A stochastic Galerkin method for first-order quasilinear hyperbolic systems with uncertainty

Xiu

2017

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This paper is concerned with generalized polynomial chaos (gPC) approximation for a general system of quasilinear hyperbolic conservation laws with uncertainty. The one-dimensional (1D) hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the Jacobian matrix. Stochastic Galerkin method is then applied to derive the equations for the gPC expansion coefficients. The resulting deterministic gPC Galerkin system is proved to be symmetrically hyperbolic. This important property then allows one to use a variety of numerical schemes for spatial and temporal discretization. Here a higher-order and path-conservative finite volume WENO scheme is adopted in space, along with a third-order total variation diminishing Runge-Kutta method in time. The method is further extended to two-dimensional (2D) quasilinear hyperbolic system with uncertainty, where the symmetric hyperbolicity of the one-dimensional system is carried over via the operator splitting technique. Several 1D and 2D numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gPC stochastic Galerkin method.

“…After those, the numerical study of the RHD equations has attracted more attention, and various exact or approximate Riemann solvers have been successively proposed, for instance the HLL (Harten-Lax-van Leer) method [31], the flux corrected transport method [8], the two-shock solvers [1,5], the Roe solver [9], the upwind scheme [10], the kinetic schemes [46,15], the flux-splitting method [7], the HLLC (HLL-Contact) scheme [25], and so forth. In addition, some higher-order accurate shock-capturing schemes were also developed for solving the RHD equations, such as ENO (essentially non-oscillatory) and weighted ENO (WENO) methods [6,49,33], finite volume local evolution Galerkin method [44], piecewise parabolic methods [20,26], adaptive mesh refinement method [50], discontinuous Galerkin (DG) method [30], direct Eulerian GRP (generalized Riemann problem) schemes [47,48,45,43], adaptive moving mesh methods [12,13], space-time conservation element and solution element method [28] and Runge-Kutta DG methods with WENO limiter [54] and so on. The readers are also referred to the review articles [21,11,22] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics

Ling

Duan

2019

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This paper studies the physical-constraints-preserving (PCP) Lagrangian finite volume schemes for one-and two-dimensional special relativistic hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the positivity of the rest-mass density and the pressure and the bound of the velocity) is proved for the first-order accurate Lagrangian scheme with the HLLC Riemann solver and forward Euler time discretization. The key is that the intermediate states in the HLLC Riemann solver are shown to be admissible or PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order accurate schemes are proposed by using the high-order accurate strong stability preserving (SSP) time discretizations and the scaling PCP limiter as well as the WENO reconstruction. Finally, several one-and two-dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the PCP Lagrangian schemes in solving the special RHD problems involving strong discontinuities, or large Lorentz factor, or low rest-mass density or low pressure etc.