S. Børve

Regularized Smoothed Particle Hydrodynamics: A New Approach to Simulating Magnetohydrodynamic Shocks

Smoothed particle hydrodynamics (SPH) has proven to be a useful numerical tool in studying a number of different astrophysical problems. Still, used on other problems, such as the modeling of low-β MHD systems, the method has so far not performed as well as one might have hoped. The present work has been motivated by the desire to accurately model strong hydrodynamic and magnetohydrodynamic shocks, and a key issue has therefore been to achieve a near-optimal representation of the simulated system at all times. Using SPH, this means combining the Lagrangian nature of the method with a smoothing-length profile that varies in both space and time. In this paper, a scheme containing two novel features is proposed. First, the scheme assumes a piecewise constant smoothing-length profile. To avoid substantial errors near the steps in the profile, alternative forms of the SPH equations of motion are used. Second, a predictive attitude toward optimizing the particle distribution is introduced by activating a mass, momentum, and energy conservation regularization process at intervals. The scheme described has been implemented in a new code called regularized smoothed particle hydrodynamics (RSPH), and test results for a number of standard hydrodynamic and magnetohydrodynamic tests in one and two dimensions using this code are presented.

SPH in spherical and cylindrical coordinates

New kernel functions for spherically, planar and cylindrically symmetric problems are developed, based on the fundamental interpolation theory of SPH. The Lagrangian formalism is used to derive the corresponding set of modified SPH equations of motion. The results show good agreement both with analytical and numerical results, in the case of the Sod shock tube test, the Noh infinite shock problem, and the Sedov point explosion test. The formulation has also been included in a 3D cylindrically symmetric problem of two colliding spherical shocks. For this latter problem, the results are presented allowing both a constant and a variable resolution. The results clearly demonstrate the capability of the new formulation to solve the singularity problem at the symmetry axis.

TWO-DIMENSIONAL MHD SMOOTHED PARTICLE HYDRODYNAMICS STABILITY ANALYSIS

Smoothed particle hydrodynamics (SPH) is an N-body integration scheme widely used within the field of astrophysics. Unfortunately, the method has up until recently been facing serious problems concerning instabilities when applied to MHD problems. Regularized smoothed particle hydrodynamics (RSPH) was proposed as an extension to SPH with the aim of achieving high-accuracy modeling of hydrodynamic and magnetohydrodynamic problems. This work included a new formulation of the discrete MHD equations that is easily implemented into SPH and RSPH codes alike. In this paper, the stability properties of two-dimensional linear MHD waves using this formulation are investigated. The presented analysis shows that linear stability properties similar to that obtained for sound waves in the absence of a magnetic field is achieved also for MHD waves. This result is confirmed by the included test results using both standard SPH and RSPH.

Multidimensional MHD shock tests of regularized smoothed particle hydrodynamics

This paper investigates to what extent the numerical scheme regularized smoothed particle hydrodynamics (RSPH) is able to accurately describe multidimensional MHD shocks. The scheme can be viewed as an extension to smoothed particle hydrodynamics (SPH), which is widely used for astrophysical applications. In the first of two previous papers, the basic idea behind the RSPH scheme was introduced and tested, primarily on one-dimensional MHD shock problems. A new formulation of the momentum equation was also proposed to secure stability in the low-β regime. A two-dimensional, linear stability analysis of this formulation was presented in the second paper. The second paper also utilized recent developments of the RSPH scheme that improve the overall description of multidimensional problems in general. Based on the results from the linear stability analysis, adjustments to the momentum equation are made in the present work, which are also applicable to the nonlinear regime. These adjustments address the problem of asymmetries in the momentum equation, which in nonlinear problems can lead to small, yet systematic errors in postshock conditions. In addition, this paper describes the first application of the improved RSPH scheme to multidimensional MHD shocks. Comparisons are made with existing methods, in particular the related SPH method. Special attention is given to the schemes ability to maintain the ∇ = 0 constraint and to what extent redefining the particle distribution affects the conservation of kinetic energy and angular momentum.

Ion phase-space vortices in 2.5-dimensional simulations

The formation and propagation of ion phase-space vortices are observed in a numerical particle-in-cell simulation in two spatial dimensions and with three velocity components. The code allows for an externally applied magnetic field. The electrons are assumed to be isothermally Boltzmann-distributed at all times, implying that Poissons equation becomes nonlinear for the present problem. Ion phase-space vortices are formed by the nonlinear saturation of the ion-ion two-stream instability, excited by injecting an ion beam at the plasma boundary. We consider the effect of a finite beam diameter and a magnetic field, in particular. A vortex instability is observed, appearing as a transverse modulation, which slowly increases with time and ultimately breaks up the vortex. When many vortices are present at the same time, we find that it is their interaction that eventually leads to a gradual filling-up of the phase-space structures. The ion phase-space vortices have a finite lifetime, which is noticeably shorter than that found in one-dimensional simulations. An externally imposed magnetic field can increase this lifetime considerably. For high injected beam velocities in magnetized plasmas, we observe the excitation of electrostatic ion-cyclotron instabilities, but see no associated formation of ion phase-space vortices. The results are relevant, for instance, for the interpretation of observations by instrumented spacecraft in the Earths ionosphere and magnetosphere.

Numerical Dissipation in RSPH Simulations of Astrophysical Flows with Application to Protoplanetary Disks

Smoothed Particle Hydrodynamics (SPH) is widely used for astrophysical applications, in particular problems of self-gravitational hydrodynamics. However, critics have argued that inherent accuracy problems with the method can be identified, in particular when it comes to describing shocks and dynamical instabilities. Regularized Smoothed Particle Hydrodynamics (RSPH) has previously been proposed as an extension to SPH. It is an attempt to increase the accuracy of the hydrodynamical description without having to abandon the Lagrangian formulation altogether. As the name implies, the method relies on a regularization technique where the solution at temporal intervals is mapped on to a new set of regularly placed particles. This technique allows us to reduce the numerical noise otherwise caused by highly irregular particle distributions and to take advantage of a more flexible approach to variable resolution. The cost of introducing the regularization scheme lies in increased methodical complexity, and in increased numerical dissipation. This paper investigates the numerical dissipation both qualitatively and quantitatively in the context of two-dimensional models relevant to the study of protoplanetary disks. Basic hydrodynamical tests highlight key properties of the RSPH approach. By comparison with an analytical solution, we are also able to quantify the dependence of the spurious viscosity on key numerical parameters. To put the theoretical discussion in perspective, we also present results from simulations of test problems involving disk-planet interactions. The results are compared to published results obtained with other codes.

Kinetic instabilities associated with injection of a plasma beam into a neutral background

Kinetic effects associated with a neutralized plasma beam injected into a neutral background are investigated by numerical methods. We assume that the dominant interactions between ions and neutrals are charge exchange collisions. It is readily noted that by these processes, a fast ion is replaced by a slow one, implying that the injected fast-ion population will generate a new slow one. For a wide parameter range, we find that the combination of these two populations will become unstable with respect to kinetic ion-sound instabilities. The resulting enhanced level of electrostatic fluctuations gives rise to an enhanced scattering of the injected ions, with a resulting very wide velocity distribution. These observations have relevance for instance for chemical processes in the plasma state, where details in the velocity distribution of the constituents can be important. The basic features of the processes outlined here are illustrated by results from a particle-in-cell simulation in one spatial dimension. The results have a wider range of applicability than those outlined here, i.e. similar observations can be made for elastic ion neutral collisions as well.

Low frequency oscillations of the magnetosphere

Sudden pulses in the model solar wind sets the Earths magnetosphere into damped oscillatory motions. A simple model is capable of explaining many of the basic properties of these oscillations, giving scaling laws for their characteristics in terms of the parameters of the problem, such as the Solar wind momentum density. The period of the oscillations, their damping and anharmonic nature are accounted for. The model has no free adjustable numerical parameters and can be seen as an effort to predict some dynamic properties of the magnetosphere on the basis of measurable steady state characteristics. A simple test of the model is found by comparing its prediction of the Earth-magnetopause distance with observed values. The results agree well with satellite observations and also numerical simulations.

Modelling high explosives (HE) using smoothed particle hydrodynamics

In this paper we present results from numerical simulations of high explosives, using a constant volume method and an axis-symmetric Regularized Smoothed Particle Hydrodynamics method. smoothed particle hydrodynamics method Empirical and numerical results show satisfactory agreement for 1 kg of detonating TNT charge. The method is further challenged with the study of shock propagation and shock reflection in complex geometries. The shock reflection pattern is altered by introducing barriers of different shapes. The effect of such barrier structures are studied.

Blast type shock wave phenomena simulated using regularized smoothed particle hydrodynamics

The numerical method RSPH is used to simulate shock reflection phenomena. In the first case we look at a shock colliding with a complex wedge configuration, whereas in the second case, the wedge is replaced by a square box. Our numerical results are found to be in good agreement with published experimental data. Reflection pattern and the pressure force on the box is studied for both short and long duration shock waves. The results demonstrate the capabilites of RSPH to handle shock reflection phenomena.