The smoothed particle hydrodynamics method is applied to an ADM 3 + 1 formulation of the equations for relativistic fluid flow. In particular the one-dimensional shock tube is addressed. Three codes are described. The first is a straightforward extension of classic SPH, while the other two are modifications which allow for time-dependent smoothing lengths. The first of these modifications approximates the internal energy density, while the second approximates the total energy density. Two smoothing forms are tested: an artificial viscosity and the direct method of A.J. Baker IFinite Element Computation Fluid Mechanics (Hemisphere, New York, 1983)]. The results indicate that the classic SPH code with particle-particle based artificial viscosity is reasonably accurate and very consistent, It gives quite sharp edges and flat plateaus, but the velocity plateau is significantly overestimated, and an oscillation can appear in the rarefaction wave. The modified versions with Baker smoothing procedure better results for moderate initial conditions, but begin to show spikes when the initial density jump is large. Generally the results are comparable to simple finite element and finite difference methods. P.J. Mann / Relativistic hydrodynamics tested with shock tube
2. Basic equations
Geometric units in which the speed of light c is set to unity are used throughout this paper. Greek indices take values from 0 to 3.
The fluid is described by the energy-momentum tensor
T"~= (p + p)u"u' 3 +pg'~,