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The paper develops high-order physical-constraint-preserving (PCP) methodsfor general relativistic hydrodynamic (GRHD) equations, equipped with a generalequation of state. Here the physical constraints, describing the admissiblestates of GRHD, are referred to the subluminal constraint on the fluid velocityand 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 andnumerical instability. To overcome the difficulties arising from the inherentstrong nonlinearity contained in the constraints, we derive an equivalentdefinition of the admissible states. Using this definition, we prove theconvexity, scaling invariance and Lax-Friedrichs (LxF) splitting property ofthe admissible state set $\mathcal G$, and discover the dependence of $\mathcalG$ on the spacetime metric. Unfortunately, such dependence yields thenon-equivalence of $\mathcal G$ at different points in curved spacetime, andinvalidates the convexity of $\mathcal G$ in analyzing PCP schemes. Thisobstacle is effectively overcame by introducing a new formulation of the GRHDequations. Based on this formulation and the above theories, a first-order LxFscheme is designed on general unstructured mesh and rigorously proved to be PCPunder a CFL condition. With two types of PCP limiting procedures, we designhigh-order, {\em provably} (not probably) PCP methods under discretization onthe proposed new formulation. These high-order methods include the PCP finitedifference, finite volume and discontinuous Galerkin methods.

Design of Provably Physical-Constraint-Preserving Methods for General Relativistic Hydrodynamics