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Eulerian smoothed particle hydrodynamics (Eulerian SPH) is considered as apotential meshless alternative to a traditional Eulerian mesh-based method,i.e. finite volume method (FVM), in computational fluid dynamics (CFD).  While researchers have analyzed the differences between these two methods,  a rigorous comparison of their performance and computational efficiency ishindered  by the constraint related to the normal direction of interfaces in pairwiseparticle interactions within Eulerian SPH framework.  To address this constraint and improve numerical accuracy,  we introduce Eulerian SPH extensions,  including particle relaxation to satisfy zero-order consistency,  kernel correction matrix to ensure first-order consistency and release theconstraint associated with the normal direction of interfaces,  as well as dissipation limiters to enhance numerical accuracy  and these extensions make Eulerian SPH rigorously equivalent to FVM.  Furthermore,  we implement mesh-based FVM within SPHinXsys, an open-source SPH library,  through developing a parser to extract necessary information from the meshfile  which is exported in the MESH format using the commercial software ICEM.  Therefore, these comprehensive approaches enable a rigorous comparisonbetween these two methods.

Extended Eulerian SPH and its realization of FVM