Glimm's Method for Relativistic Hydrodynamics

TL;DR

This paper compares Glimm's method to finite differencing methods in relativistic hydrodynamics, showing it excels at resolving sharp features like shocks but is less effective for smooth flows.

Contribution

It demonstrates the advantages and limitations of Glimm's method in relativistic hydrodynamics test problems, highlighting its superior shock resolution.

Findings

Glimm's method yields smaller errors for shock profiles.

Standard methods perform better for smooth flows.

Glimm's method is useful for problems with sharp discontinuities.

Abstract

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.