Computer codes containing both hydrodynamics and radiation play a central role in simulating both astrophysical and inertial confinement fusion phenomena. A crucial aspect of these codes is the necessity of solving the radiation diffusion equations implicitly in the presence of hydrodynamic shocks, turbulence, and a 4-5 order of magnitude variation in an opacity that is also spatially anisotropic. We present the results of a comparison of five different linear solvers (diagonally scaled conjugate gradient, incomplete LU preconditioner with GMRES, incomplete Cholesky with conjugate gradient, multigrid, and multigrid preconditioned conjugate gradient) on a variety of radiation and radiation-hydrodynamic problems. Each problem is run using each solver over three ranges of scale (small [1,000 zones], medium [10,000 zones], and large [100,000 zones]). In addition, all problems are run on an ALE (Arbitrary Lagrangian Eulerian) mesh that responds dynamically to fluid flow. We find that for the pure radiation flow problems that multigrid is a clear winner as problem size increases. The algorithm is scalable over problem size. We also find that there is a close correlation between solver performance and time step controls in the code. For problems involving hydrodynamic flow, incomplete Cholesky with conjugate gradient is comparable to multigrid due to the fact that the time step is severely restricted by the Courant condition.This in turn means the matrix tends to stay weakly diagonally dominant. This statement is true over all problem sizes. Therefore, the cheaper cost of incomplete Cholesky with conjugate gradient means that even though this method requires more iterations to converge per cycle than multigrid, it compares favorably with multigrid primarily because of the fact that the latter requires fewer iterations to converge but at a greater cost per iteration. We expect that the multigrid would again be a clear winner over all solvers in radiation- hydrodynamic problems if the problem size continued to increase or less severe time step controls were required.