A multignd solver was applied to the simple 1-D Boltzmann equation [21] for the linear case with no electron-electron collisions. The findings from this were compared to those of an existing direct solver for the same problem The Boltzmann problem was defined for a N2 plasma. A basic Gauss-Seidel iteration was found to act as the best smoother for the multigrid solver. The Galerkin method of restriction was found to work, while the direct method failed Reasons for this are suggested. An adaptive method was developed which may slightly improve performance m some cases. It was found for 256 points that multigrid only used 20% of the Work Units required by the direct solver Given that the direct LU-decomposition solver requires V3N3 manipulations, and the multigrid Vcycle uses ON^logN, the method has increasing advantage for larger systems. The efficiency also improves with increasing dimension Another important advantage of multigrid is that there should be no considerable loss of efficiency when solving the non linear case which includes electron-electron collisions.