Two‐dimensional MHD Smoothed Particle Hydrodynamics Stability Analysis

Børve, S.; Omang, M.; Trulsen, J.

doi:10.1086/421520

Cited by 31 publications

(49 citation statements)

References 14 publications

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“…The stability properties of the M n kernel family have been investigated by a number of authors (Morris 1996;Børve et al 2004;Read et al 2010;Dehnen & Aly 2012, V12). A crucial result which emerges from these analyses is that all of the B-splines suffer from pairing instability.…”

Section: Kernelsmentioning

confidence: 99%

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

Valdarnini

2016

ApJ

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In this paper we present results from a series of hydrodynamical tests aimed at validating the performance of a smoothed particle hydrodynamics (SPH) formulation in which gradients are derived from an integral approach. We specifically investigate the code behavior with subsonic flows, where it is well known that zeroth-order inconsistencies present in standard SPH make it particularly problematic to correctly model the fluid dynamics.In particular we consider the Gresho-Chan vortex problem, the growth of Kelvin-Helmholtz instabilities, the statistics of driven subsonic turbulence and the cold Keplerian disc problem. We compare simulation results for the different tests with those obtained, for the same initial conditions, using standard SPH. We also compare the results with the corresponding ones obtained previously with other numerical methods, such as codes based on a moving-mesh scheme or Godunov-type Lagrangian meshless methods.We quantify code performances by introducing error norms and spectral properties of the particle distribution, in a way similar to what was done in other works. We find that the new SPH formulation exhibits strongly reduced gradient errors and outperforms standard SPH in all of the tests considered. In fact, in terms of accuracy we find good agreement between the simulation results of the new scheme and those produced using other recently proposed numerical schemes. These findings suggest that the proposed method can be successfully applied for many astrophysical problems in which the presence of subsonic flows previously limited the use of SPH, with the new scheme now being competitive in these regimes with other numerical methods.

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Section: Kernelsmentioning

confidence: 99%

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

Valdarnini

2016

ApJ

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“…The stability properties of SPH have been investigated by a number of authors (Swegle & Hicks 1995;Morris 1996;Monaghan 2000;Børve et al 2004;Read et al 2010;Dehnen & Aly 2012). The instabilities are studied analytically by analyzing the dispersion relation for sound waves of small amplitude, propagating in a uniform medium.…”

Section: Stability Issuesmentioning

confidence: 99%

Hydrodynamic capabilities of an SPH code incorporating an artificial conductivity term with a gravity-based signal velocity

Valdarnini

2012

A&A

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This paper investigates the hydrodynamic performances of an smoothed particle hydrodynamics (SPH) code incorporating an artificial heat conductivity term in which the adopted signal velocity is applicable when gravity is present. To this end, we analyze results from simulations produced using a suite of standard hydrodynamical test problems. In accordance with previous findings, we show that the performances of SPH in describing the development of Kelvin-Helmholtz instabilities depend strongly on both the consistency of the initial condition set-up and the leading error in the momentum equation due to incomplete kernel sampling. In contrast, the presence of artificial conductivity does not significantly affect the results. An error and stability analysis shows that the quartic B-spline kernel (M 5 ) possesses very good stability properties and so we propose its use with a large neighbor number, between ∼50 (2D) to ∼100 (3D), to improve convergence in simulation results without being affected by the so-called clumping instability. Moreover, the results of the Sod shock tube demonstrate that to obtain simulation profiles in accord with the analytic solution, for simulations employing kernels with a non-zero first derivative at the origin, it is necessary to use a much larger number of neighbors than in the case of the M 5 runs. Our SPH simulations of the blob test show that in order to achieve blob disruption it is necessary to include an artificial conductivity term. However, we find that in the regime of strong supersonic flows an appropriate limiting condition, which depends on the Prandtl number, must be imposed on the artificial conductivity SPH coefficients in order to avoid an unphysical amount of heat diffusion. Our results from hydrodynamic simulations that include self-gravity show profiles of hydrodynamic variables that are in much better agreement with those produced using mesh-based codes. In particular, the final levels of core entropies in cosmological simulations of galaxy clusters are consistent with those found using AMR codes. This demonstrates that the proposed diffusion scheme is capable of mimicking the process of entropy mixing that is produced during structure formation because of the diffusion caused by turbulence. Finally, the results of our Rayleigh-Taylor instability test demonstrate that in the regime of very subsonic flows the code still has several difficulties in the treatment of hydrodynamic instabilities. These problems are intrinsic to the way in which standard SPH gradients are calculated and not to the implementation of the artificial conductivity term. To overcome these difficulties, several numerical schemes have been proposed that, if coupled with the SPH implementation presented in this paper, could solve the issues that have recently been addressed in investigating SPH performances to model subsonic turbulence.

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“…Alternatively one can use the stable MHD force formulated by Børve et al (2004), that is, where the source term…”

Section: Hybrid Approachmentioning

confidence: 99%

“…Price & Bate (2007, 2008. The flaw in this methodology is that, since the equations of motion are not derived with the constraint of the numerical formulation of the induction equation, there is no guarantee that total energy will be conserved 3 (and indeed, using either the Morris force or the Børve et al (2004) approach, total momentum conservation is not guaranteed either, though the errors are quite small even for shock-type problems, see Price 2004). In this context total energy conservation means that…”

Section: Hybrid Approachmentioning

confidence: 99%

“…Whilst the proof that this correction term indeed stabilises the standard SPMHD equations has been given by Børve et al (2004), it is unclear why it should also work for the vector potential formulation. Empirically, we find that subtracting the B(∇ · B) term can indeed stabilise the vector potential force, but only in a limited range of circumstances, the limitations of which are unclear.…”

Section: Instability Correctionmentioning

confidence: 99%

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Smoothed Particle Magnetohydrodynamics: Some Shocking Results

Price¹,

Monaghan

2004

Astrophysics and Space Science

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In this paper we investigate the use of the vector potential as a means of maintaining the divergence constraint in the numerical solution of the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. We derive a self-consistent formulation of the equations of motion using a variational principle that is constrained by the numerical formulation of both the induction equation and the curl operator used to obtain the magnetic field, which guarantees exact and simultaneous conservation of momentum, energy and entropy in the numerical scheme. This leads to a novel formulation of the MHD force term, unique to the vector potential, which differs from previous formulations. We also demonstrate how dissipative terms can be correctly formulated for the vector potential such that the contribution to the entropy is positive definite and the total energy is conserved.On a standard suite of numerical tests in one, two and three dimensions we find firstly that the consistent formulation of the vector potential equations is unstable to the well-known SPH tensile instability, even more so than in the standard Smoothed Particle Magnetohydrodynamics (SPMHD) formulation where the magnetic field is evolved directly. Furthermore we find that, whilst a hybrid approach based on the vector potential evolution equation coupled with a standard force term gives good results for one and two dimensional problems (where dA z /dt = 0), such an approach suffers from numerical instability in three dimensions related to the unconstrained evolution of vector potential components. We conclude that use of the vector potential is not a viable approach for Smoothed Particle Magnetohydrodynamics.

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Two‐dimensional MHD Smoothed Particle Hydrodynamics Stability Analysis

Cited by 31 publications

References 14 publications

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

Improved Performances in Subsonic Flows of an SPH Scheme With Gradients Estimated Using an Integral Approach

Hydrodynamic capabilities of an SPH code incorporating an artificial conductivity term with a gravity-based signal velocity

Smoothed Particle Magnetohydrodynamics: Some Shocking Results

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