CppCon 2024

Could we revisit numerical integrators in the light of C++26 and bring more genericity, performance, and expressivity to the domain? In this talk, we will explore how modern C++ techniques can add something new and relevant to one of the oldest and most basic task of scientific computing: the integration of systems of equations. We will examine, in particular, how most numerical integrators can be derived from a small set of first principles that can be easily mapped onto C++ concepts and composable algorithmic building blocks. One of the goal of the approach introduced in this presentation will be to achieve as much as possible with the simplest and smallest amount of code. C++23 and C++26 programming techniques, including reflection, will be leveraged to transfer some of the burden of implementation to the compiler while still ensuring maximum performance.

In practice, the talk will combine aspects of high-performance computing, numerical methods, and software architecture. We will start by summarizing recent discoveries made in applied mathematics on Runge-Kutta methods, linear multistep methods, and general linear methods to see how it can help design better abstractions that can be translated into C++ concepts. We will then examine how a few carefully crafted algorithmic building blocks can be combined to generate the whole diversity of numerical integrators from first principles. The automation of this approach using C++23 and C++26's reflection to make the compiler generate highly efficient code will be then discussed in great length. Next, we will dive into parallelization strategies, including distributed ones as well as heterogenous computing. Some perspectives will also be given on the possibility for the compiler to branch on the best integrator given the mathematical properties of a system of equations as well as ways to derive new integrators at compile-time.

To illustrate our approach, we will examine the behavior and performance of numerous integrators on several real-world problems including a supercomputing N-body code for cosmology that simulates the gravitational dynamics of large scale astrophysical structures in an expanding Universe. A great care will be taken to make all the code and examples as reproducible and standalone as possible so that most of the presented content can simply be copied and pasted to make it work everywhere. Finally, even if the talk will focus on the particular case of numerical integration, the methodology presented in this talk will be applicable everywhere in scientific computing and beyond to achieve better software architecture in technical contexts.