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 × 595 × 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.

Abstract Computational fluid dynamics (CFD) simulations have emerged as a powerful tool for understanding the multiphase flows that occur in a wide range of engineering applications and natural processes. A multiphase CFD code called Multiphase Flow with Interphase eXchanges (MFIX) has been under development at the National Energy Technology Laboratory (NETL) since the 1980s for modeling the multiphase flows that occur in fossil fuel reactors. CFD codes such as MFIX are equipped with a number of numerical algorithms to solve a large set of coupled partial differential equations over three‐dimensional grids consisting of hundreds of thousands of cells on parallel computers. Currently the next‐generation version of MFIX is under development with the goal of building a multiphase problem‐solving environment (PSE) that would facilitate the simple reuse of modern software components by application scientists. Several open‐source frameworks were evaluated to identify the best‐suited framework for the multiphase PSE. There are many requirements for the multiphase PSE and each of these open‐source frameworks offers functionalities that satisfy the requirements to varying extents. Therefore, matching the requirements and the functionalities is not a simple task and requires a systematic and quantitative decision‐making procedure. We present a multi‐criteria decision‐making approach for determining a major system design decision and demonstrate its application on the framework‐selection problem. Copyright © 2006 John Wiley & Sons, Ltd.