The use of a high-level, abstract

The abstract mathematical theory of partial differential equations (PDEs) is formulated in terms of manifolds, scalar fields, tensors, and the like, but these algebraic structures are hardly recognizable in actual PDE solvers. The general aim of the Sophus programming style is to bridge the gap between theory and practice in the domain of PDE solvers. Its main ingredients are a library of abstract datatypes corresponding to the algebraic structures used in the mathematical theory and an algebraic expression style similar to the expression style used in the mathematical theory. Because of its emphasis on abstract datatypes, Sophus is most naturally combined with object-oriented languages or other languages supporting abstract datatypes. The resulting source code patterns are beyond the scope of current compiler optimizations, but are sufficiently specific for a dedicated source-to-source optimizer. The limited, domain-specific, character of Sophus is the key to success here. This kind of optimization has been tested on computationally intensive Sophus style code with promising results. The general approach may be useful for other styles and in other application domains as well.

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It has long been acknowledged that the development of scientific applications is in need of better software engineering practices. Here we contrast the difference between conventional software development of CFD codes with a method based on coordinate free mathematics. The former approach leads to programs where different aspects, such as the discretisation technique and the coordinate systems, can get entangled with the solver algorithm. The latter approach yields programs that segregate these concerns into fully independent software modules. Such considerations are important for the construction of numerical codes for practical problems. The two approaches are illustrated on the coating problem: the simulation of coating a wire with a polymer.

This paper presents ideas for using coordinate-free numerics in modern Fortran to achieve code flexibility in the partial differential equation (PDE) domain. We also show how Fortran, over the last few decades, has changed to become a language well-suited for state-of-the-art software development. Fortran’s new coarray distributed data structure, the language’s class mechanism, and its side-effect-free, pure procedure capability provide the scaffolding on which we implement HPC software. These features empower compilers to organize parallel computations with efficient communication. We present some programming patterns that support asynchronous evaluation of expressions comprised of parallel operations on distributed data. We implemented these patterns using coarrays and the message passing interface (MPI). We compared the codes’ complexity and performance. The MPI code is much more complex and depends on external libraries. The MPI code on Cray hardware using the Cray compiler is 1.5–2 times faster than the coarray code on the same hardware. The Intel compiler implements coarrays atop Intel’s MPI library with the result apparently being 2–2.5 times slower than manually coded MPI despite exhibiting nearly linear scaling efficiency. As compilers mature and further improvements to coarrays comes in Fortran 2015, we expect this performance gap to narrow.

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Data parallelism has appeared as a fruitful approach to the parallelisation of compute-intensive programs. Data parallelism has the advantage of mimicking the sequential (and deterministic) structure of programs as opposed to task parallelism, where the explicit interaction of processes has to be programmed. In data parallelism data structures, typically collection classes in the form of large arrays, are distributed on the processors of the target parallel machine. Trying to extract distribution aspects from conventional code often runs into problems with a lack of uniformity in the use of the data structures and in the expression of data dependency patterns within the code. Here we propose a framework with two conceptual classes, Machine and Collection. The Machine class abstracts hardware communication and distribution properties. This gives a programmer high-level access to the important parts of the low-level architecture. The Machine class may readily be used in the implementation of a Collection class, giving the programmer full control of the parallel distribution of data, as well as allowing normal sequential implementation of this class. Any program using such a collection class will be parallelisable, without requiring any modification, by choosing between sequential and parallel versions at link time. Experiments with a commercial application, built using the Sophus library which uses this approach to parallelisation, show good parallel speed-ups, without any adaptation of the application program being needed.