Glimm's Method for Relativistic Hydrodynamics

Cannizzo, J. K.
Gehrels, N.
Vishniac, E. T.
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We present the results of standard one-dimensional test problems in relativistic hydrodynamics using Glimm's (random choice) method, and compare them to results obtained using finite differencing methods. For problems containing profiles with sharp edges, such as shocks, we find Glimm's method yields global errors ~1-3 orders of magnitude smaller than the traditional techniques. The strongest differences are seen for problems in which a shear field is superposed. For smooth flows, Glimm's method is inferior to standard methods. The location of specific features can be off by up to two grid points with respect to an exact solution in Glimm's method, and furthermore curved states are not modeled optimally since the method idealizes solutions as being composed of piecewise constant states. Thus although Glimm's method is superior at correctly resolving sharp features, especially in the presence of shear, for realistic applications in which one typically finds smooth flows plus strong gradients or discontinuities, standard FD methods yield smaller global errors. Glimm's method may prove useful in certain applications such as GRB afterglow shock propagation into a uniform medium.
Comment: 42 pages, 18 figures - to appear in the Astrophysical Journal, June 1, 2008, v. 679