Improved time scheme integration approach for dealing with semi analytical boundary conditions in SPARTACUS2D

The present paper aims to provide robust and physical boundary conditions in the most basic SPH formulation, that is without renormalization of the kernel gradient or Riemann solver approach. The present method is analogous to Kulasegaram et. als approach [1] in the sense that it is based on a geometrical parameter measuring the missing area in the kernel support when a particle is in the vicinity of a solid boundary, but improve accuracy for linear fields, such as a hydrostatic pressure. This geometrical term is computed with a governingequation, and the integration in time of the continuity equation is modified in order to ensure density conservation during longtime simulations. Results are presented for (i) a channel flow where hydrostatic pressure is better reproduced and density is sustained, (ii) still water and dam break in a basin with a wedge, (iii) the SPHERIC benchmark of a moving square solid inside a rectangular tank.