Relation between diffusive terms and Riemann solver in WCSPH

The widely used weakly compressible variant of Smoothed Particle Hydrodynamics (SPH) method suffers from density and hence pressure oscillations. This is due to the particle Lagrangian nature of the SPH method in combination with weakly compressible assumption, explicit time scheme and that the substitutions of the derivatives in the SPH method are central. There are two common strategies how to suppress these issues. One of them is to use numerical diffusive term which is added to the continuity equation in order to suppress the spurious oscillation of density field. The second option is to describe the particle -particle interaction in terms of Riemann problem and use Riemann solver, which provides numerical dissipation, to handle particle interactions. In our work, we deal with the relation between these two approaches. For the constant reconstruction and for the linear reconstruction we show that the usage of Riemann solvers is due to its intrinsic numerical viscosity equivalent to the usage of diffusive terms based on even derivatives, with the difference that the Riemann solvers lead to a significantly higher diffusivity value then the standard diffusive terms. We also discuss the usage of limiters for cases with the linear reconstruction of the solution. Moreover, for both, the constant and the linear reconstruction, we analyze additional terms resulting from the employed Riemann solver also for the momentum equation. Combining these results we obtain equivalent partial differential equations, which are the result of the usage of the Riemann solver.