The IMP is a fascinating and playful entity, known for its delightful mischief. It bestows unwavering stability upon any SPH practitioner clever enough to unravel a conundrum --a fixed-point problem that eludes even the most determined attempts of Newton's method.
The IMP offers several advantages that make it an attractive choice for SPH simulations.
One of the key benefits of the IMP method is its ability to ensure unconditional stability in SPH models. This is achieved through exact energy conservation, which eliminates the need for dissipation terms.
Currently, the energy conservation property of the IMP method has been demonstrated for compressible and Weakly-Compressible SPH formulations. Ongoing theoretical work is also being conducted on Incompressible SPH.
Although the IMP method requires solving a fixed-point problem, numerical experiments have shown that the residues converge spectrally.
An ideal scenario for numerical simulations
In fact, the IMP method has been shown to outperform commonly used time integrators in SPH, such as Improved-Euler and 4th order Runge-Kutta, with just 4 or 5 iterations.
Adopting the IMP method is relatively straightforward and requires minimal additional code.
| Code | Core code lines | Other code lines |
|---|---|---|
| AQUAgpusph | 116 | 33 |
| TNL-SPH | 314 | 0 |
| DualSPHysics | ~500 | ~10 |
| PySPH | ~750 | ~0 |
| GPUSPH | ~1000 | ~50 |
Currently, AQUAgpusph and TNL-SPH are the only IMP adopters. However, estimates are provided for other popular SPH implementations. If you are a developer who has adopetd an IMP, please submit an issue so that we can update the table.
Some details on how it is implemented on AQUAgpusph can be found here
Cercos-Pita, J. L., et al. "SPH energy conservation for fluid–solid interactions." Computer Methods in Applied Mechanics and Engineering 317 (2017): 771-791.
Cercos-Pita, J. L., et al. "The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations." European Journal of Mechanics-B/Fluids 97 (2023): 78-92. (Available on arXiv)
Cercos-Pita, J. L., et al. "Boundary conditions for SPH through energy conservation." Computers & Fluids 285 (2024): 106454. (Available on arXiv)
Merino-Alonso, P. & Violeau, D. "Energy conservation in ISPH" 17th ERCOFTAC SPHERIC workshop (2023).