Completeness, conservation and error in SPH for fluids

Abstract

SUMMARYSmoothed particle hydrodynamics (SPH) is becoming increasingly common in the numerical simulation of complex fluid flows and an understanding of the errors is necessary. Recent advances have established techniques for ensuring completeness conditions (low-order polynomials are interpolated exactly) are enforced when estimating property gradients, but the consequences on errors have not been investigated. Here, we present an expression for the error in an SPH estimate, accounting for completeness, an expression that applies to SPH generally. We revisit the derivation of the SPH equations for fluids, paying particular attention to the conservation principles. We find that a common method for enforcing completeness violates a property required of the kernel gradients, namely that gradients with respect the two position variables be equal and opposite. In such models this means conservation principles are not enforced and we present results that show this. As an aside we show the summation interpolant for density is a solution of, and may be used in the place of, the discretized, symmetrized continuity equation. Finally, we examine two examples of discretization errors, namely numerical boundary layers and the existence of crystallized states.

On the kernel and particle consistency in smoothed particle hydrodynamics

“…We first demonstrate that the normalization condition of the kernel is independent of h, suggesting that its discrete representation depends only on n, consistently with the error analyses of Vaughan et al [23] and Read et al [21]. Although C 0 and C 1 kernel and particle consistency can be achieved by some corrective SPH methods, a simple observation shows that C 2 kernel consistency is difficult to achieve, implying an upper limit to the convergence rate of SPH in practical applications.…”

“…This has important implications on the issue of particle consistency in that the discrete summation form of the integral consistency relations will only depend on the number of neighboring particles within the kernel support. While this result was previously derived from detailed SPH error analyses by Vaughan et al [23] and Read et al [21], we find as a further implication of our analysis that C 2 kernel consistency is difficult to achieve in actual SPH simulations due to an intrinsic diffusion, which is closely related to the inherent dispersion of the particle positions relative to the mean. This implies that in practical applications SPH has a limiting second-order convergence rate.…”

“…As was pointed out by Liu and Liu [12] and formally demonstrated by Vaughan [22], for a bounded domain the kernel approximation (2) needs be replaced by the integral form of the Shepard interpolant to guarantee C 0 consistency near a model boundary. Equation (9) bears some resemblance with the expression of the error derived by Vaughan et al [23], where the contribution to the error due to the smoothing length can be separated from that due to the discretized form of the integral, which, being independent of h, will directly depend on the number of neighbors within the kernel support. The SPH discretization makes reference to a set of Lagrangian particles which may, in general, be disordered.…”

Improved MLPG_R method for simulating 2D interaction between violent waves and elastic structures

“…The examples of studies related to momentum conservation in SPH methods can be seen in [33][34][35]. For the MPS method, Khayyer and Goto [36] studied the momentum conservation property and modified the MPS method to conserve the momentum equations locally.…”

Monte Carlo simulations of intensity profiles for energetic particle propagation

“…Alternatively, the following will be inspired by the smoothed particle hydrodynamics (SPH) approach (e.g., Vaughan et al 2008;Springel 2010). The basic idea is that all computational particles are no longer treated as pointlike physical particles.…”