On the Convergence of SPH Method for Scalar Conservation Laws with Boundary Conditions

Abstract

This paper is the third of a series where the convergence analysis of SPH method for multidimensional conservation laws is analyzed. In this paper, two original numerical models for the treatment of boundary conditions are elaborated. To take into account nonlinear effects in agreement with Bardos, LeRoux and Nedelec boundary conditions ([1], [14]), the state at the boundary is computed by solving appropriate Riemann problems. The first numerical model is developed around the idea of boundary forces in surrounding walls, recently initiated in [33] by Monaghan in his simulation of gravity currents. The second one extends the well-known approach of ghost particles for plane boundaries to the case of general curved boundaries. The convergence analysis in L p loc (p < ∞) is achieved thanks to the uniqueness result of measure-valued solutions recently established in [3] for L ∞ initial and boundary data.

SPH simulation of free surface flow over a sharp-crested weir

“…This has not been achieved together before by any other published SPH scheme. Indeed, the unphysical and spurious oscillations of the Gingold and Monaghan scheme [9] completely disappear in the pressure profiles even at the solid wall boundaries and the numerical diffusion does not smear the free surface profile, unlike the Ben Moussa and Vila approach [15,20]. This new method [8] applies an upwind Rusanov-type flux in the discrete form of the density equation.…”

“…Another approach has been proposed by Ben Moussa and Vila [15,20] who use Riemann solvers to evaluate the interactions between each couple of particles replacing the artificial viscosity term of Gingold and Monaghan [9] by an intrinsic numerical viscosity, automatically contained in the numerical flux without parameters to calibrate. This approach considerably improves the numerical solution of the pressure field, but unfortunately, it is too diffusive to compute the particle positions correctly and hence it cannot be used to simulate violent free surface problems, such as the flow over a sharp-crested weir.…”

From continuum mechanics to SPH particle systems and back: Systematic derivation and convergence

“…In particular, we are interested in those motion mappings that are continuous and differentiable in time, and we wish to obtain their equation of motion. The introduction of the motion mapping ( t ) t∈[0,T ] has taken us from pure Eulerian coordinates in (1) towards Lagrangian (material) coordinates in (2). The crucial and final step to complete this procedure is now to specify what the velocity field u is.…”

“…The reader should note that the notation used in [33] differs substantially from ours, but that the philosophy of deriving the equations of motion is the same. 2 If e = e(ρ, y), and moreover, we include nonconservative forces, then instead of (21) we obtain…”

A simple two-phase method for the simulation of complex free surface flows

“…More accurate and more stable SPH algorithms for the computation of free surface flows have been recently presented in [86,62,19,35,33], to name just a few. For a rigorous analysis of SPH methods, see the work of Vila [85] and Ben Moussa [55].…”