A multi-rate time integration scheme for smoothed particle hydrodynamics

We present a general multi-rate (MR) time integration scheme for smoothed particle hydrodynamics (SPH) that enhances computational efficiency while preserving stability and accuracy. Unlike conventional single-rate methods, which enforce the most restrictive time step globally, the proposed approach splits the time advancement operator into substeps pertaining to advection, acoustic, viscous, and thermal effects, and advances each by considering its characteristic time scale. Validation against canonical benchmarks, including the Taylor–Green vortex, a three-dimensional dambreak, thermal Couette flow, and porous media flow, demonstrates that the scheme reproduces analytical solutions and experimental reference data with accuracy equal to that of single-rate integration. At the same time, it significantly reduces computational costs, with speed-ups up to 4× depending on the degree of time-scale difference. The implementation is straightforward. We employ a modular, graphics processing unit (GPU)-accelerated framework. The approach is particularly advantageous for multi-physics problems, where time scales differ widely. This work establishes multi-rate time integration as a robust and efficient alternative to single-rate (SR) schemes for advancing the capabilities of SPH in engineering and scientific applications.