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This paper first studies the admissible state set $\mathcal G$ ofrelativistic magnetohydrodynamics (RMHD). It paves a way for developingphysical-constraints-preserving (PCP) schemes for RMHD equations with thesolutions in $\mathcal G$. To overcome the difficulties arising from theextremely strong nonlinearities and no explicit formulas of the primitivevariables and flux vectors with respect to the conservative vector, twoequivalent forms of $\mathcal G$ with explicit constraints on the conservativevector are skillfully discovered. The first is derived by analyzing roots ofseveral polynomials and transferring successively them, and further used toprove the convexity of $\mathcal G$ with the aid of semi-positive definitenessof the second fundamental form of a hypersurface. While the second is derivedbased on the convexity and then used to show the orthogonal invariance of$\mathcal G$. The Lax-Friedrichs (LxF) splitting property does not holdgenerally for nonzero magnetic field, but by a constructive inequality andpivotal techniques, we discover the generalized LxF splitting properties,combining the convex combination of some LxF splitting terms with a discretedivergence-free condition of magnetic field. Based on above analyses, severalone- and two-dimensional PCP schemes are then studied. In 1D case, afirst-order accurate LxF type scheme is first proved to be PCP under the CFLcondition, and then high-order PCP schemes are proposed via a PCP limiter. In2D case, the discrete divergence-free condition and PCP property are analyzedfor a first-order LxF scheme, and two sufficient conditions are derived forhigh-order accurate PCP schemes. Our analysis reveals in theory for the firsttime that the discrete divergence-free condition is closely connected with thePCP property. Several numerical examples demonstrate theoretical findings andperformance of numerical schemes.

Admissible state and physical constraints preserving schemes for relativistic magnetohydrodynamic equations