Relativistic Hydrodynamics 2025, 13 - Formulations of the Hydrodynamic Equations
The Wilson-formulation of the hydrodynamic equations takes the transport velocity to define basic dynamical variables D = ρut, Sμ = ρhutuμ, E = ρ utε Retaining conservation or rest mass, energy and momentum will create a system of equations referred to as the "Wilson formulation of the equations of relativistic hydrodynamics".
In general, first-order forms of hydrodynamic equations follow the schema ∂t U + A ⋅ ∇U + B = 0 which is hyperbolic if A is diagonalisable with a set of real eigenspeeds and linearly independent eigenvectors. Hyperbolic equations are numerically well-solvable, so with A(U) the Jacobian of a flux vector F(U), then the conservative form of the system is ∂tU + ∇F(U) = 0 with the vector of conserved variables U. If the system allows discontinuities in its solution, it's more convenient to have an integral formulation to have the discontinuities embedded in the mathematics. It solves then for a weak solution of the conservative equation, which itself needs to satisfy the weak formulation.
The 3+1 Valencia formulation is applied to high-resolution shock-capturing solutions, centered on the identity ∇μ(ρuμ) = 0 ↠ D = ραut = ρW, ∂tD = 0 This allows the quantities in the vector of conserved quantities U to be solved, as long as their equations are valid in any curved spacetime written in any coordinate system with appropriate line element. Linear combinations of conserved variables are still solutions of the equations in conservative form.
A covariant formulation would be invariant wrt. spacetime foliation, so its derivation is straightforward from the conservation equations. In short, it reduces to ∂μF^μ = ∂tFt + ∂i Fi = S Both the Valencia and the covariant formulation contain source terms on the right side that vanish only in flat spacetime. In the covariant formulation, no partial derivatives of the fluid variables exist to make things difficult for conservation considerations. It does however not have the gauge information available from the typical 3 + 1 decomposition, nor is the vector of conserved variables easily invertible.