Various methods of simulating diffusion phenomena with parallel hardware are discussed. In particular methods are compared requiring local and global communication among the processors in terms of total computation time. Systolic convolution on a locally connected array is seen to exhibit an asymptotic advantage over Fourier methods on a globally connected array. Whereas this may translate into a numerical advantage for extremely large numbers of ultrafast devices for two-dimensional systems, this is unlikely for three-dimensional systems. Thus global Fourier methods will be advantageous for three-dimensional systems for foreseeable device speeds and system sizes. The fact that optical interconnections are potentially advantageous for implementing the longer connections of such globally connected systems suggests that they can be beneficially employed in future nanoelectronic computers. Heat removal considerations play an important role in our conclusions.