We present a series of 2-dimensional hydrodynamic simulations of massive disks around protostars .
We simulate the same physical problem using both a ‘ Piecewise Parabolic Method ’ ( PPM ) code and a ‘ Smoothed Particle Hydrodynamic ’ ( SPH ) code , and analyze their differences .

The disks studied here range in mass from 0.05 M _ { * } to 1.0 M _ { * } and in initial minimum Toomre Q value from 1.1 to 3.0 .
We adopt simple power laws for the initial density and temperature in the disk with an isothermal ( \gamma = 1 ) equation of state .
The disks are locally isothermal .
We allow the central star to move freely in response to growing perturbations .
The simulations using each code are compared to discover differences due to error in the methods used .
For this problem , the strengths of the codes overlap only in a limited fashion , but similarities exist in their predictions , including spiral arm pattern speeds and morphological features .
Our results represent limiting cases ( i.e .
systems evolved isothermally ) rather than true physical systems .

Disks become active from the inner regions outward .
From the earliest times , their evolution is a strongly dynamic process rather than a smooth progression toward eventual nonlinear behavior .
Processes that occur in both the extreme inner and outer radial regions affect the growth of instabilities over the entire disk .
Effects important for the global morphology of the system can originate at quite small distances from the star .
We calculate approximate growth rates for the spiral patterns ; the one-armed ( m = 1 ) spiral arm is not the fastest growing pattern of most disks .
Nonetheless , it plays a significant role due to factors which can excite it more quickly than other patterns .
A marked change in the character of spiral structure occurs with varying disk mass .
Low mass disks form filimentary spiral structures with many arms while high mass disks form grand design spiral structures with few arms .

In our SPH simulations , disks with initial minimum Q = 1.5 or lower break up into proto-binary or proto-planetary clumps .
However , these simulations can not follow the physics important for the flow and must be terminated before the system has completely evolved .
At their termination , PPM simulations with similar initial conditions show uneven mass distributions within spiral arms , suggesting that clumping behavior might result if they were carried further .
Simulations of tori , for which SPH and PPM are directly comparable , do show clumping in both codes .
Concern that the point-like nature of SPH exaggerates clumping , that our representation of the gravitational potential in PPM is too coarse , and that our physics assumptions are too simple , suggest caution in interpretation of the clumping in both the disk and torus simulations .