Special Relativistic Smoothed Particle Hydrodynamics Based on Riemann Solver

TL;DR

The paper addresses accurate simulation of special-relativistic hydrodynamics with SPH and introduces SRGSPH, a Godunov-based SPH framework that leverages a Riemann solver for inter-particle fluxes and convolution-based field estimates to enhance shock capturing. It also proposes a volume-based density definition and a variable smoothing-length scheme to handle nonuniform baryon numbers while preserving conservation properties. The method is validated through a comprehensive suite of one- and two-dimensional shock-tube problems and Kelvin-Helmholtz instability tests, demonstrating robust accuracy and smooth handling of density variations. The results indicate that SRGSPH offers improved resolution of shocks and discontinuities in relativistic flows, with potential extensions to general-relativistic hydrodynamics for high-energy astrophysical applications.

Abstract

This paper proposes a novel numerical method based on Godunov Smoothed Particle Hydrodynamics for special relativistic fluid dynamics. Our method utilizes a Riemann solver to describe shock, enhancing accuracy in strong shock waves. The formulation maintains conservation laws and achieves higher accuracy through convolution integrals that define physical quantities for SPH particles. We also propose the number density calculation method that uses a non-equal baryon number in each SPH particle and variable smoothing length in a way different from the conventional method. Numerical experiments demonstrate the method's robustness across one- and two-dimensional relativistic shock tube problems, as well as its ability to simulate Kelvin-Helmholtz instabilities accurately, validating SRGSPH as a reliable approach for high-resolution relativistic simulations.

Figures (15)

• Figure 1: The spatial profiles of the baryon number, smoothing length, and number density profile in the lab frame for each SPH particle in (a) gather and (b) scatter approaches. As field quantities, the density in the lab frame is represented by the dashed line and the smoothing length by the dotted line. The gather method defines the density in the lab frame and smoothing length as field quantities, whereas the scatter method defines only the density in the lab frame as a field quantity and does not specify the smoothing length as a field quantity.
• Figure 2: The spatial profiles of the baryon number, smoothing length, and number density profile in the lab frame for each SPH particle under the volume-based and the standard approaches.
• Figure 3: The spatial profiles of the baryon number, smoothing length, and number density in the lab frame for each SPH particle in a general case where both the baryon number and smoothing length vary spatially. In the volume-based approach, the smoothing length varies smoothly across the interface, in contrast to the standard approach, where it exhibits a discontinuous jump.
• Figure 4: The result of the Sod problem at $t = 0.35$. The symbols represent the numerical solutions and the solid lines show the analytical solutions. For this problem, each SPH particle has the same baryon number. The left-hand side contains 3200 SPH particles and the right-hand side contains 400. Calculated with $\gamma_c = 5/3$.
• Figure 5: Results of the Sod problem at $t = 0.35$ with different baryon numbers across the initial discontinuity. Both panels show numerical solutions (symbols) compared with the analytical solutions (solid lines). The left panel corresponds to the volume-based approach, and the right panel to the standard approach. The number of SPH particles on each side of the discontinuity is 1800. We set $\gamma_c = 5/3$.