This article describes the porting and optimization of an explicit, time-dependent, computational fluid dynamics code on an 8,192-node MasPar MP-1. The MasPar is a very fine-grained, single instruction, multiple data parallel computer. The code uses the flux-corrected transport algorithm. We describe the techniques used to port and optimize the code, and the behavior of a test problem. The test problem used to benchmark the flux-corrected transport code on the MasPar was a two-dimensional exploding shock with periodic boundary conditions. We discuss the performance that our code achieved on the MasPar, and compare its performance on the MasPar with its performance on other architectures. The comparisons show that the performance of the code on the MasPar is slightly better than on a CRAY Y-MP for a functionally equivalent, optimized two-dimensional code.