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The relativistic hydrodynamics (RHD) equations have three crucial intrinsicphysical constraints on the primitive variables: positivity of pressure anddensity, and subluminal fluid velocity. However, numerical simulations canviolate these constraints, leading to nonphysical results or even simulationfailure. Designing genuinely physical-constraint-preserving (PCP) schemes isvery difficult, as the primitive variables cannot be explicitly reformulatedusing conservative variables due to relativistic effects. In this paper, wepropose three efficient Newton--Raphson (NR) methods for robustly recoveringprimitive variables from conservative variables. Importantly, we rigorouslyprove that these NR methods are always convergent and PCP, meaning theypreserve the physical constraints throughout the NR iterations. The discoveryof these robust NR methods and their PCP convergence analyses are highlynontrivial and technical. As an application, we apply the proposed NR methodsto design PCP finite volume Hermite weighted essentially non-oscillatory(HWENO) schemes for solving the RHD equations. Our PCP HWENO schemesincorporate high-order HWENO reconstruction, a PCP limiter, andstrong-stability-preserving time discretization. We rigorously prove the PCPproperty of the fully discrete schemes using convex decomposition techniques.Moreover, we suggest the characteristic decomposition with rescaledeigenvectors and scale-invariant nonlinear weights to enhance the performanceof the HWENO schemes in simulating large-scale RHD problems. Several demandingnumerical tests are conducted to demonstrate the robustness, accuracy, and highresolution of the proposed PCP HWENO schemes and to validate the efficiency ofour NR methods.

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