A Density Independent Formulation of Smoothed Particle Hydrodynamics

Abstract: The standard formulation of the smoothed particle hydrodynamics (SPH) assumes that the local density distribution is differentiable. This assumption is used to derive the spatial derivatives of other quantities. However, this assumption breaks down at the contact discontinuity. At the contact discontinuity, the density of the low-density side is overestimated while that of the high-density side is underestimated. As a result, the pressure of the low (high) density side is over (under) estimated. Thus, unphysical repulsive force appears at the contact discontinuity, resulting in the effective surface tension. This tension suppresses fluid instabilities. In this paper, we present a new formulation of SPH, which does not require the differentiability of density. Instead of the mass density, we adopt the internal energy density (pressure), and its arbitrary function, which are smoothed quantities at the contact discontinuity, as the volume element used for the kernel integration. We call this new formulation density independent SPH (DISPH). It handles the contact discontinuity without numerical problems. The results of standard tests such as the shock tube, Kelvin-Helmholtz and Rayleigh-Taylor instabilities, point like explosion, and blob tests are all very favorable to DISPH. We conclude that DISPH solved most of known difficulties of the standard SPH, without introducing additional numerical diffusion or breaking the exact force symmetry or energy conservation. Our new SPH includes the formulation proposed by Ritchie & Thomas (2001) as a special case. Our formulation can be extended to handle a non-ideal gas easily.

• The paper introduces a pressure-based SPH formulation that overcomes errors at contact discontinuities.
• It validates the method with shock tube, Kelvin-Helmholtz, and Rayleigh-Taylor tests, demonstrating enhanced accuracy over traditional SPH.
• The authors show that the novel DISPH framework effectively simulates complex fluid dynamics and holds potential for extension to non-ideal gases and magnetohydrodynamics.

An Analysis of a Density Independent Formulation of Smoothed Particle Hydrodynamics

The paper by Saitoh and Makino introduces a novel approach to Smoothed Particle Hydrodynamics (SPH) referred to as Density Independent Smoothed Particle Hydrodynamics (DISPH). The standard formulation of SPH, while widely used, suffers from significant limitations, particularly at the interface of contact discontinuities. This paper addresses these deficiencies by proposing a formulation that does not rely on the differentiability of density, instead utilizing internal energy density (pressure) as the primary smoothing scale for kernel integration.

Key Contributions

The authors identify that the existing SPH framework inaccurately evaluates physical quantities at contact discontinuities due to the assumptions regarding density differentiability. This leads to notable errors in pressure estimation across different density regions, often resulting in unphysical forces that suppress natural fluid instabilities like the Kelvin-Helmholtz instability. By recasting the smoothing mechanism in terms of pressure rather than density, DISPH aims to mitigate these inaccuracies and improve the physical realism of interfacial simulations.

Experimental Validation

The paper substantiates the effectiveness of DISPH through a series of standard test cases:

• Shock Tube Tests: The pressure distribution, especially near contact discontinuities, is significantly more accurate in DISPH than in traditional SPH, avoiding the artificial repulsive forces.
• Kelvin-Helmholtz and Rayleigh-Taylor Instabilities: DISPH demonstrates enhanced accuracy in capturing these instabilities due to the smoothed and consistent pressure gradients maintained across discontinuities.
• Point Impact and Blob Tests: These further corroborate the capability of DISPH to handle complex fluid structures and to preserve accurate dynamics even in the presence of high-density contrasts.

Implications and Theoretical Enhancement

The adoption of pressure-based smoothing addresses critical issues regarding the contact discontinuity handling in SPH simulations. This reformulation not only improves upon the existing Lagrangian schemes in computational astrophysics but has potential applications in engineering where hydrodynamical stability at phase interfaces is critical. Theoretical advancements stemming from this work include the possibility of extending this framework to non-ideal gases, further augmenting the versatility of the SPH method.

Future Directions

Future studies could expand upon this density-independent approach by evaluating its performance on more complex fluid dynamics simulations, including those involving phase changes and compressible flows. Moreover, extending DISPH to incorporate additional physics models, such as magnetohydrodynamics, could broaden its applicability in both astrophysical and engineering domains.

In summary, the formulation proposed by Saitoh and Makino represents a substantial step forward in SPH methodology, directly addressing long-standing challenges related to density discontinuities and providing a pathway for more accurate and stable fluid simulations. This work opens new avenues not only for SPH but also for other particle-based methods where accurate interfacial dynamics are crucial.