• gravitation;
  • hydrodynamics;
  • relativistic processes;
  • methods: numerical


We develop a numerical hydrodynamics code using a pseudo-Newtonian formulation that uses the weak-field approximation for the geometry, and a generalized source term for the Poisson equation that takes into account relativistic effects. The code was designed to treat moderately relativistic systems such as rapidly rotating neutron stars. The hydrodynamic equations are solved using a finite volume method with high-resolution shock-capturing techniques. We implement several different slope limiters for second-order reconstruction schemes and also investigate higher order reconstructions such as the piecewise parabolic method, essentially non-oscillatory method (ENO) and weighted ENO. We use the method of lines to convert the mixed spatial-time partial differential equations into ordinary differential equations (ODEs) that depend only on time. These ODEs are solved using second- and third-order Runge–Kutta methods. The Poisson equation for the gravitational potential is solved with a multigrid method, and to simplify the boundary condition, we use compactified coordinates which map spatial infinity to a finite computational coordinate using a tangent function. In order to confirm the validity of our code, we carry out four different tests including one- and two-dimensional shock tube tests, stationary star tests of both non-rotating and rotating models, and radial oscillation mode tests for spherical stars. In the shock tube tests, the code shows good agreement with analytic solutions which include shocks, rarefaction waves and contact discontinuities. The code is found to be stable and accurate: for example, when solving a stationary stellar model the fractional changes in the maximum density, total mass, and total angular momentum per dynamical time are found to be 3 × 10−6, 5 × 10−7 and 2 × 10−6, respectively. We also find that the frequencies of the radial modes obtained by the numerical simulation of the steady-state star agree very well with those obtained by linear analysis.