The partial differential Eqs [2.1, 2.2, and 2.3], along with the equation of state 2.4, which describe the time evolution of compressible fluid flow can be solved without the use of a Lagrangian mesh. The method follows embedded fluid points and uses finite difference approximations to \(\vec \nabla\)P and \(\vec \nabla\) · \(\vec u\) to update p, \(\vec u\) and e. We have demonstrated that the method can accurately calculate highly distorted flows without difficulty. The finite difference approximations are not unique, improvements may be found in the near future. The neighbor selection is not unique, but the one being used at present appears to do an excellent job. The method could be directly extended to three dimensions. One drawback to the method is the failure toexplicitly conserve mass, momentum and energy. In fact, at any given time, the mass is not defined. We must perform an auxiliary calculation by integrating the density field over space to obtain mass, energy and momentum. However, in all cases where we have done this, we have found the drift in these quantities to be no more than a few percent.