We describe a modified form of Smoothed Particle Hydrodynamics (SPH) in which the specific thermal energy equation is based on a compatibly differenced formalism, guaranteeing exact conservation of the total energy. We compare the errors and convergence rates of the standard and compatible SPH formalisms on analytic test problems involving shocks. We find that the new compatible formalism reliably achieves the expected first-order convergence in such tests, and in all cases improves the accuracy of the numerical solution over the standard formalism.
Smoothed Particle Hydrodynamics (SPH) is a meshless or particle based Lagrangian approach to modeling hydrodynamics. SPH was originally developed as a method for studying astrophysical systems ( [5], [3]), for which purpose it has a number of advantages: it's particle-like nature is well suited for combination with existing N-body gravitational techniques; the Lagrangian frame of SPH naturally follows the large dynamic range of length and mass scales that gravitationally unstable systems undergo; the lack of an imposed geometry from an underlying mesh suits the shifting, complex, three-dimensional nature of astrophysical objects. These properties have made SPH modeling a successful tool for studying galaxy formation and evolution, star formation, the formation and evolution of planetary bodies, etc. SPH has also found many applications outside of astrophysics as well, such as modeling of metal casting, material fracture, impact of bodies into water, and multi-phase flows, among others.
The standard SPH formalism (outlined in [6] & [1]) time evolves the specific thermal energy in a manner that does not conserve total energy, and in recent years it has been found that the energy conservation with this approach can become quite poor ( [4],
Email address: mikeowen@llnl.gov (J. Michael Owen).
[8], [15]). These authors suggest a few modifications on the standard SPH formalism to improve the energy conservation: [8] and [12] find that accounting for the variability of the SPH smoothing scale in the SPH dynamical equations improves the consistency and thereby the energy conservation of the scheme; in [15] the authors choose to evolve the specific entropy equation in place of the energy equation (including a term which also accounts for the variable smoothing scale) and likewise find that overall energy conservation is improved. None of these approaches guarantees energy conservation, however, typically improving the total conservation to ∼1% on tests cited.
In this paper we adopt a different approach to this problem, and formulate an exactly energy preserving form of SPH based on the concept of compatible differencing outlined in [2]. The resulting form of SPH is manifestly energy preserving (to machine roundoff) by construction, irrespective of details such as the presence of a variable smoothing scale. This paper is layed out as follows: §2 describes the standard SPH formalism we will be comparing with; §3 derives the new compatible method for evolving the SPH specific thermal energy; §4 demonstrates the performance of the new scheme on several test problems of interest; finally §5 discusses our conclusions.
Before we begin, a word about notation. In this paper we use the convention that Latin subscripts denote node indices (m i is the mass of node i), while Greek superscripts denote dimensional indices (x α i is the αth component of the position for node i). We use the summation convention for repeated Greek indices:
For conciseness we represent the spatial gradient as ∂ α F ≡ ∂F/∂x α .
The SPH dynamical equations describing the evolution of the mass, momentum, and energy have been derived elsewhere in detail (see for instance [6] and [1]), so we will simply summarize the forms we are using:
where for a given node i ρ i is the mass density, m i the mass, v α i the velocity, P i the pressure, and u i the specific thermal energy. W and ∂ α W represent the interpolation kernel and it’s gradient -the form of W we use is the cubic B-spline, given in [6]. Because W (r, h) is a function of distance and the smoothing scale, the subscripts on W i = W (r, h i ) and W j = W (r, h j ) denote which nodes definition of the smoothing scale has been used. The terms with both i and j indices indicate either differences (v
The term Π αβ ij is the artificial viscosity, expressed here appropriately for a tensor viscosity as described in [11]. The majority of tests presented in this paper are one-dimensional, in which case Π αβ ij is equivalent to the well known scalar Monaghan-Gingold form of Π ij described in [6] and [1]. It is only in the two-dimensional cylindrical Noh problem presented in §4.3 that the results using the tensor viscosity are distinct from those using the normal scalar form of the viscosity. This is required because otherwise the errors imposed by the standard viscosity dominate the solution for this problem, masking the effects we seek to study in this work. SPH using Eqs. ( 1) -( 3) has a number of useful properties. Conservation of total mass is ensured since the mass is always simply the sum of the masses of the nodes. Because Eq. ( 2) is symmetric for each interacting pair of nodes i and j, the total linear momentum is guaranteed to be conserved because pair-wise forces are always equal and opposite. So long as the pair-wise forces are also radially aligned between points, conservation of angular momentum is also ensured. However, if non-radial forces are used (such as is the case with the tensor vi
This content is AI-processed based on open access ArXiv data.