Numerical simulation of the spherically symmetric Einstein-Euler(EE)system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inheren...Numerical simulation of the spherically symmetric Einstein-Euler(EE)system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric evolution.This paper proposes a high-order constraint-preserving(CP)compact oscillation-eliminating discontinuous Galerkin(cOEDG)method specifically tailored to address these difficulties.The method integrates a scaleinvariant oscillation-eliminating mechanism[M.Peng,Z.Sun,K.Wu,Math.Comp.,94(2025)]into a compact Runge-Kutta DG framework.By characterizing the convex invariant region of the hydrodynamic subsystem with general barotropic equations of state,we prove that the proposed scheme preserves physical realizability(specifically,positive density and subluminal velocity)directly in terms of conservative variables,thereby eliminating the need for complex primitive-variable checks.To ensure the geometric validity of the spacetime,we introduce a bijective transformation of the metric potentials.Rather than evolving the constrained metric components directly,the scheme advances unconstrained auxiliary variables whose inverse mapping automatically enforces strict positivity and asymptotic bounds without any limiters.The resulting CP compact OEDG(CPcOEDG)method exhibits robustness and the expected accuracy in capturing strong gravity-fluid interactions,as demonstrated by simulations of black hole accretion and relativistic shock waves.展开更多
Numerical approximation of the Ericksen-Leslie system with variable density is considered in this paper.The spherical constraint condition of the orientation field is preserved by using polar coordinates to reformulat...Numerical approximation of the Ericksen-Leslie system with variable density is considered in this paper.The spherical constraint condition of the orientation field is preserved by using polar coordinates to reformulate the system.The equivalent new system is computationally cheaper because the vector function of the orientation field is replaced by a scalar function.An iteration penalty method is applied to construct a numerical scheme so that stability is improved.We first prove that the scheme is uniquely solvable and unconditionally stable in energy.Then we show that this scheme is of first-order convergence rate by rigorous error estimation.Finally,some numerical simulations are performed to illustrate the accuracy and effectiveness of the scheme.展开更多
基金supported by the Science Challenge Project(Grant No.TZ2025007)the National Natural Science Foundation of China(Grant No.124B2022)+1 种基金the Shenzhen Science and Technology Program(Grant No.RCJC20221008092757098)the Guangdong Basic and Applied Basic Research Foundation(Grant No.2024A1515012329).
文摘Numerical simulation of the spherically symmetric Einstein-Euler(EE)system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric evolution.This paper proposes a high-order constraint-preserving(CP)compact oscillation-eliminating discontinuous Galerkin(cOEDG)method specifically tailored to address these difficulties.The method integrates a scaleinvariant oscillation-eliminating mechanism[M.Peng,Z.Sun,K.Wu,Math.Comp.,94(2025)]into a compact Runge-Kutta DG framework.By characterizing the convex invariant region of the hydrodynamic subsystem with general barotropic equations of state,we prove that the proposed scheme preserves physical realizability(specifically,positive density and subluminal velocity)directly in terms of conservative variables,thereby eliminating the need for complex primitive-variable checks.To ensure the geometric validity of the spacetime,we introduce a bijective transformation of the metric potentials.Rather than evolving the constrained metric components directly,the scheme advances unconstrained auxiliary variables whose inverse mapping automatically enforces strict positivity and asymptotic bounds without any limiters.The resulting CP compact OEDG(CPcOEDG)method exhibits robustness and the expected accuracy in capturing strong gravity-fluid interactions,as demonstrated by simulations of black hole accretion and relativistic shock waves.
基金supported by Research Project Supported by Shanxi Scholarship Council of China(2021-029)Shanxi Provincial International Cooperation Base and Platform Project(202104041101019)Shanxi Province Natural Science Foundation(202203021211129).
文摘Numerical approximation of the Ericksen-Leslie system with variable density is considered in this paper.The spherical constraint condition of the orientation field is preserved by using polar coordinates to reformulate the system.The equivalent new system is computationally cheaper because the vector function of the orientation field is replaced by a scalar function.An iteration penalty method is applied to construct a numerical scheme so that stability is improved.We first prove that the scheme is uniquely solvable and unconditionally stable in energy.Then we show that this scheme is of first-order convergence rate by rigorous error estimation.Finally,some numerical simulations are performed to illustrate the accuracy and effectiveness of the scheme.
