Relativistic Glimm’s method

3.2 Relativistic Glimm’s method

Wen et al. [295

] have extended Glimm’s random choice method [104 ] to 1D SRHD. They developed a first-order accurate hydrodynamic code combining Glimm’s method (using an exact Riemann solver) with standard finite difference schemes.

In the random choice method, given two adjacent states $n uj$ and $n uj+1$ at time $n t$ , the value of the numerical solution at time $tn+1∕2$ and position $xj+1∕2$ is given by the exact solution $u(x,t)$ of the Riemann problem evaluated at a randomly chosen point inside zone (j, j + 1), i.e.,

where $ξn$ is a random number in the interval [0, 1].

Besides being conservative on average, the main advantages of Glimm’s method are that it produces both completely sharp shocks and contact discontinuities, and that it is free of diffusion and dispersion errors.

Chorin [52 ] applied Glimm’s method to the numerical solution of homogeneous hyperbolic conservation laws. Colella [57 ] proposed an accurate procedure of randomly sampling the solution of local Riemann problems, and investigated the extension of Glimm’s method to two dimensions using operator splitting methods.