AbstractAn algorithm is presented that solves a linear advection‐diffusion problem using a least‐squares formulation and a conjugate gradient method to solve the corresponding minimization problem. An implementation in CM‐Fortran on a Thinking Machines CM‐2 is compared with a serial implementation on an IBM RS6000. The maximum speed‐up obtained is a factor of 70. For fine grids, the CPU time scales almost ideally when the number of processors is increased from 409