Special issue: Advanced stencil‐code engineering

Stencil codes are compute-intensive algorithms, in which data points arranged in a large grid are being recomputed repeatedly from the values of data points in a predefined neighborhood. This fixed neighborhood pattern is called a stencil. Stencil codes see wide-spread use in computing the discrete solutions of partial differential equations and systems composed of such equations. Connected to the implementation of stencil codes is the use of efficient solver technology, ie, iterative solvers that rely on the application of a stencil and that provide good convergence properties like multigrid methods. Major application areas are the natural sciences and engineering. Although, in many of these applications, unstructured adaptive discretizations support an efficient use of supercomputers whose architectures possibly include accelerators or are of a heterogeneous nature, the use of structured discretizations and, thus, stencil codes has turned out to be helpful. Stencil codes are algorithms with a pleasantly high regularity: The data structures are higher-dimensional grids, and the computations follow a static, locally contained dependence pattern and are typically arranged in nested loops with linearly affine bounds. This invites massive parallelism and raises the hope for easily achieved high performance. However, serious challenges remain: • Because of the large numbers and varieties of stencil code implementations, deriving each of them individually, even if by code modification from one another, is not practical. Not even the use of program libraries is practical; instead, a domain-specific metaprogramming approach is needed. • Reaching petascale to exascale execution speed is a challenge in the frequently used so-called multigrid algorithms, which work on a hierarchy of increasingly larger grids. The coarse grids in the upper part of the hierarchy are too small for massive parallelism. • Efficiency, ie, a high ratio of speedup to the degree of parallelism, is impaired by the low mathematica density, ie, the low ratio of computation steps to data transfers of stencil codes. • An inappropriate use of the execution platform may act as a performance brake.