Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

Cai，Chaoyi1; Qiu，Jianxian2; Wu，Kailiang3

2024-02-01

10.1016/j.jcp.2023.112669

Journal of Computational Physics   影响因子和分区

0021-9991

1090-2716

The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton–Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. Our NR methods are versatile and can be seamlessly incorporated into any RHD schemes that require the recovery of primitive variables. As an application, we apply them to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods.

Finite volume scheme Hermite WENO High-order accuracy Newton–Raphson method Physical-constraint-preserving Relativistic hydrodynamics

SCI ; EI

英语

通讯

PHYSICS

2-s2.0-85178101654

Scopus

1.School of Mathematical Sciences,Xiamen University,Xiamen,Fujian,361005,China
2.School of Mathematical Sciences,Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing,Xiamen University,Xiamen,Fujian,361005,China
3.Department of Mathematics,SUSTech International Center for Mathematics,Southern University of Science and Technology,National Center for Applied Mathematics Shenzhen (NCAMS),Shenzhen,518055,China

Cai，Chaoyi,Qiu，Jianxian,Wu，Kailiang. Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics[J]. Journal of Computational Physics,2024,498.

Cai，Chaoyi,Qiu，Jianxian,&Wu，Kailiang.(2024).Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics.Journal of Computational Physics,498.

Cai，Chaoyi,et al."Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics".Journal of Computational Physics 498(2024).