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The numerical solution of many problems in continuum dynamics is seriously limited by the computation rates attainable on computers with serial architecture. Parallel processing machines can achieve much higher rates. However, applying additional processors to a calculation is only part of the solution. In this report, parallel algorithms are developed for explicit and implicit, Lagrangian and Eulerian finite difference schemes for computational continuum dynamics in one spatial dimension. First, the explicit conservation equations in the Lagrangian reference frame are readily reformulated for concurrent processing. Second, and implicit solution is derived for these equations. This is important because it yields unconditional stability. The parallelism is achieved via a block implicit numerical scheme. Third, a rezoning algorithm is employed with each Lagrangian integration step to transform the mesh back to the Eulerian reference frame. Along the algorithmic development path, a zone-by-zone parallelization gives way to a block-by-block technique both of which are self-scheduling. Then the latter is compared to an approach that keeps the parallel processes alive for many time steps. At each step of this research exploiting the architectural advantages of the HEP H1000 (Heterogeneous Element Processor) computer. (Author)