Improved versions of MPS method have been proposed for enhancement of numerical stability (e.g. [https://archive.today/20130105131748/http://www3.interscience.wiley.com/journal/2910/abstract Koshizuka et al., 1998]; [https://dx.doi.org/10.1002/fld.1106 Zhang et al., 2005]; [https://dx.doi.org/10.1016/j.fluiddyn.2005.12.002 Ataie-Ashtiani and Farhadi, 2006];[https://dx.doi.org/10.1002/fld.2132 Shakibaeina and Jin, 2009]; [https://doi.org/10.1016/j.cma.2019.112771 Jandaghian and Shakibaeinia, 2020]), momentum conservation (e.g. Hamiltonian MPS by [https://dx.doi.org/10.1016/j.cma.2006.12.006 Suzuki et al., 2007]; Corrected MPS by [https://dx.doi.org/10.1142/S0578563408001788 Khayyer and Gotoh, 2008]; Enhanced MPS by [https://doi.org/10.1016/j.cma.2019.112771 Jandaghian and Shakibaeinia, 2020]), mechanical energy conservation (e.g. Hamiltonian MPS by [https://dx.doi.org/10.1016/j.cma.2006.12.006 Suzuki et al., 2007]), pressure calculation (e.g. [https://dx.doi.org/10.1016/j.coastaleng.2008.10.004 Khayyer and Gotoh, 2009], [https://dx.doi.org/10.1002/fld.2207 Kondo and Koshizuka, 2010], [https://dx.doi.org/10.1016/j.apor.2010.01.001 Khayyer and Gotoh, 2010], [https://doi.org/10.1016/j.compfluid.2019.104235 Xu and Jin, 2019]), and for simulation of multiphase and granular flows ([http://ascelibrary.org/doi/10.1061/%28ASCE%29HY.1943-7900.0001275 Nabian and Farhadi 2016]; [https://doi.org/10.1017/jfm.2021.320 Xu and Jin, 2021]; [https://doi.org/10.1007/s11440-022-01766-4 Xu and Li, 2022]).