SPHyNX: An Open-Source C++ Library for Solving Stochastic Differential Equations using Smoothed Particle Hydrodynamics

This research concerns the development of a numerical code for simulating the evolution of the Probability Density Function (PDF) of stochastic processes. Each stochastic process is first defined using a traditional Stochastic Differential Equation (SDE), which is then transformed into a corresponding equation for the evolution of the PDF. This equation is known as the Fokker-Planck-Kolmogorov (FPK) equation, which is a nonlinear advection-diffusion differential equation. To solve the FPK equation, the Smoothed Particle Hydrodynamics (SPH) method is employed, leading to a mesh-free Lagrangian approach, widely used in fluid dynamics. This choice is motivated by the fact that the SPH intrinsically imposes requirements such as conservation of mass and positivity of the PDF. The whole FPK-SPH framework is implemented in C++ programming language, with a focus on high-performance computing through multi-core parallelization. The FPK-SPH method is then validated against several test cases, namely stochastic versions of the Duffing oscillator, Lorenz system and SIRD epidemiological model. This includes comparisons with other numerical methods in terms of accuracy, robustness, and computational efficiency. Overall, the developed FPK-SPH method is shown to be a complementary approach to traditional Monte Carlo (MC) simulations, as the former directly provides the PDF evolution, while the latter relies on statistical inference from a large number of realizations. Ultimately, the combination of an SPH approach to solving the FPK equation and its implementation in an open C++ framework is a practical tool for uncertainty quantification in stochastic systems, with potential applications in countless fields such as physics, engineering, finance, and biology.