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Particle hydrodynamics with accurate gradients: a comparison of different formulations

S. Rosswog

TL;DR

This study evaluates how gradient accuracy in Smoothed Particle Hydrodynamics (SPH) affects hydrodynamic performance across shocks and instabilities. It compares five SPH formulations that differ in gradient prescriptions (standard kernel gradients, linearly exact LE gradients, and reproducing kernel gradients) and examines two dissipation strategies: shock-based dissipation and Roe's approximate Riemann solver, including a noise-triggered dissipation mechanism. The key finding is that gradient accuracy is crucial for instability growth, with reproducing kernel gradients (V3) delivering the best overall results and linearly exact gradients (V2) offering a close, computationally cheaper alternative; Roe-based Riemann solution (V4) excels in shocks but can suppress instabilities, while aLE-based gradients (V2) strike a favorable balance between accuracy and cost. The results guide practical SPH development, suggesting that gradient-accurate formulations with slope-limited dissipation provide robust, high-fidelity performance for instabilities and complex flows.

Abstract

We compare here several modern versions of SPH with a particular focus on the impact of gradient accuracy. We examine specifically an approximation to the "linearly exact" gradients (aLE) with standard SPH kernel gradients and with linearly reproducing kernels (RPKs) that fulfill the lowest order consistency relations exactly by construction. Most of the explored SPH formulations use shock dissipation (i.e. artificial viscosity and conductivity) with slope-limited reconstruction and parameters that trigger on both shocks and noise. We also compare with a recent particle hydrodynamics formulation that uses both RPKs and Roe's approximate Riemann solver instead of shock dissipation. Not too surprisingly, we find that the shock tests are rather insensitive to the gradient accuracy, but whenever instabilities are involved the gradient accuracy plays a crucial role. The reproducing kernel gradients perform best, but they are closely followed by the much simpler aLE gradients. The Riemann solver approach has some (minor) advantages in the shock tests, but shows some resistance against instability growth and at low resolution the corresponding Kelvin-Helmholtz simulations show substantially slower instability growth than the best shock dissipation approaches. Based on the battery of benchmark tests performed here, we consider a shock dissipation approach with reproducing kernels (our versions $V_3$ and $V_5$) as best, but closely followed by a similar version ($V_2$) that uses the simpler and computationally cheaper aLE gradients.

Particle hydrodynamics with accurate gradients: a comparison of different formulations

TL;DR

This study evaluates how gradient accuracy in Smoothed Particle Hydrodynamics (SPH) affects hydrodynamic performance across shocks and instabilities. It compares five SPH formulations that differ in gradient prescriptions (standard kernel gradients, linearly exact LE gradients, and reproducing kernel gradients) and examines two dissipation strategies: shock-based dissipation and Roe's approximate Riemann solver, including a noise-triggered dissipation mechanism. The key finding is that gradient accuracy is crucial for instability growth, with reproducing kernel gradients (V3) delivering the best overall results and linearly exact gradients (V2) offering a close, computationally cheaper alternative; Roe-based Riemann solution (V4) excels in shocks but can suppress instabilities, while aLE-based gradients (V2) strike a favorable balance between accuracy and cost. The results guide practical SPH development, suggesting that gradient-accurate formulations with slope-limited dissipation provide robust, high-fidelity performance for instabilities and complex flows.

Abstract

We compare here several modern versions of SPH with a particular focus on the impact of gradient accuracy. We examine specifically an approximation to the "linearly exact" gradients (aLE) with standard SPH kernel gradients and with linearly reproducing kernels (RPKs) that fulfill the lowest order consistency relations exactly by construction. Most of the explored SPH formulations use shock dissipation (i.e. artificial viscosity and conductivity) with slope-limited reconstruction and parameters that trigger on both shocks and noise. We also compare with a recent particle hydrodynamics formulation that uses both RPKs and Roe's approximate Riemann solver instead of shock dissipation. Not too surprisingly, we find that the shock tests are rather insensitive to the gradient accuracy, but whenever instabilities are involved the gradient accuracy plays a crucial role. The reproducing kernel gradients perform best, but they are closely followed by the much simpler aLE gradients. The Riemann solver approach has some (minor) advantages in the shock tests, but shows some resistance against instability growth and at low resolution the corresponding Kelvin-Helmholtz simulations show substantially slower instability growth than the best shock dissipation approaches. Based on the battery of benchmark tests performed here, we consider a shock dissipation approach with reproducing kernels (our versions and ) as best, but closely followed by a similar version () that uses the simpler and computationally cheaper aLE gradients.
Paper Structure (28 sections, 76 equations, 16 figures, 1 table)

This paper contains 28 sections, 76 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. SPH approximation
  4. Gradients in SPH
  5. Linearly exact gradients and their approximations
  6. Reproducing kernel gradients
  7. Measuring gradient accuracy
  8. Shock dissipation
  9. Controlling dissipation
  10. SPH formulations
  11. Version 0 ($V_0$): "traditional SPH" with cubic spline kernels and constant dissipation
  12. Version 1 ($V_1$): $\Psi=\rho$ with standard kernel gradients
  13. Version 2 ($V_2$): $\Psi=\rho$ with aLE-gradients
  14. Version 3 ($V_3$): $\Psi=\rho$ with reproducing kernel gradients
  15. Version 4 ($V_4$): reproducing kernels and approximate Riemann solver
  16. ...and 13 more sections

Figures (16)

  • Figure 1: Numerical experiment to measure gradient accuracy. The upper row shows the particle positions, exactly arranged on a cubic lattice (left), and the same lattice, but with each particle coordinate shifted by a small random number chosen uniformly from the interval [-0.001,0.001]. The second row shows the gradient errors for both particle configurations for the different gradient gradient prescription. Note that the hardly visible randomization of the positions leads to a serious gradient deterioration for the standard, the aLE- and the IA-gradient. Please see the main text for the exact definitions of the gradient estimates.
  • Figure 2: Density in a Sedov explosion at $t=0.9$ for all SPH versions.
  • Figure 3: 3D Riemann problem 1 for all SPH versions at $t=0.2$
  • Figure 4: 3D Riemann problem 2 for all SPH versions at $t=0.2$
  • Figure 5: Kelvin-Helmholtz instability for all SPH versions at $512^2$.
  • ...and 11 more figures