  The performance of CPU-based and GPU-based systems is often low for PDE
codes, where large, sparse, and often structured systems of linear equations
must be solved. Iterative solvers are limited by data movement, both between
caches and memory and between nodes. Here we describe the solution of such
systems of equations on the Cerebras Systems CS-1, a wafer-scale processor that
has the memory bandwidth and communication latency to perform well. We achieve
0.86 PFLOPS on a single wafer-scale system for the solution by BiCGStab of a
linear system arising from a 7-point finite difference stencil on a 600 X 595 X
1536 mesh, achieving about one third of the machine's peak performance. We
explain the system, its architecture and programming, and its performance on
this problem and related problems. We discuss issues of memory capacity and
floating point precision. We outline plans to extend this work towards full
applications.
