Shock-capturing particle hydrodynamics with reproducing kernels

TL;DR

The paper tackles accurate shock-capturing in particle-based hydrodynamics by integrating Roe's approximate Riemann solver with slope-limited reconstruction within a linearly reproducing kernel (RPK) SPH framework. By enforcing zeroth- and first-order consistency through the RPKs, the method reproduces constant and linear fields to machine precision and uses $(\nabla \mathcal{W})_{ab}$ gradients in the final equations. The approach is validated across 3D benchmarks including spherical and Sedov blasts, Kelvin–Helmholtz and Rayleigh–Taylor instabilities, and Schulz–Rinne-type shocks, showing excellent agreement with analytic/reference solutions and robust performance across limiters (minmod, van Albada, vanLeerMC), with van Albada recommended. The work advances SPH by achieving reduced artificial dissipation while maintaining accurate shock and instability dynamics, enabling more reliable simulations in astrophysical and fluid-structure contexts. Overall, the combined Roe-RPK framework demonstrates strong potential for precise, low-dissipation, shock-capturing particle hydrodynamics in 3D systems.

Abstract

We present and explore a new shock-capturing particle hydrodynamics approach. Our starting point is a commonly used discretization of smoothed particle hydrodynamics. We enhance this discretization with Roe's approximate Riemann solver, we identify its dissipative terms, and in these terms, we use slope-limited linear reconstruction. All gradients needed for our method are calculated with linearly reproducing kernels that are constructed to enforce the two lowest-order consistency relations. We scrutinize our reproducing kernel implementation carefully on a "glass-like" particle distribution, and we find that constant and linear functions are recovered to machine precision. We probe our method in a series of challenging 3D benchmark problems ranging from shocks over instabilities to Schulz-Rinne-type vorticity-creating shocks. All of our simulations show excellent agreement with analytic/reference solutions.

Figures (21)

• Figure 1: Illustration of one of SPH's most salient features: the natural treatment of vacuum. The plot shows how two stars have been ripped apart by a black hole (lurking at the coordinate origin) into spaghetti-like, thin gas streams that are held together by the gas' self-gravity. The inset shows the initial conditions of the initially spherical stars. The original stars in the inset only cover a fraction of $10^{-9}$ of the volume that is shown at late times (t= 371 h). Simulation performed with the MAGMA2 code, see Rosswog (2020).
• Figure 2: To add dissipation, Riemann problems are solved for each particle and its neighbour particles.
• Figure 3: The particle distribution that is used to scrutinize the linear reproducing kernel method. The 3D particle distribution has been set up using a Centroidal Voronoi Tessellation; shown is a slice with a thickness of $|z| < 6 \times 10^{-3}$.
• Figure 4: Approximation of function value (left) and derivative in $x$-direction (right), the exact result is in both cases $=1$. Each time we show the SPH-approximation in black, the reproducing kernel (RPK-) result in red.
• Figure 5: Logarithm of the function error (left) and of the error in the $x$-derivative approximation (right). Each time we show the SPH-approximation in black and the reproducing kernel (RPK-) result in red.