MHD Schemes Research Articles

A fourth-order accurate finite volume scheme for resistive relativistic MHD

A fourth-order accurate finite volume scheme for resistive relativistic MHD

A New Flux Splitting Method and Central Difference Scheme for Magnetohydrodynamics

In this work, we describe a new flux splitting method for an ideal MHD in the 3D curvilinear coordinate system. The method is designed to create a dissipation term by zero flux term of flux split and create a new central difference scheme for ideal MHD. Numerical tests are performed to demonstrate the accuracy and capability of the new scheme to capture complex waves of ideal MHD.

Finite element scheme for MHD forced convection flow near stagnation point and heat transfer by Newtonian heating, constant wall temperature and constant heat flux

Two dimensional, steady, forced convection magnetohydrodynamic flow of an incompressible, viscous electrically conducting fluid in a forward stagnation region of an infinite solid surface with Newtonian heating, constant wall temperature and constant heat flux has been investigated. Governing partial differential equations for the exploration have been formulated and converted to nonlinear ordinary differential equations by inserting convenient variables. An efficient finite element scheme along to Gauss elimination method has been introduced to find the numerical solutions of the resultant equations. Variation in velocity and temperature distributions against the pertinent parameters like magnetic parameter, Prandtl number and Eckert number have been displayed graphically while skin-friction coefficient and Nusselt number have been discussed quantitatively. A comparison of the computational results has been found in excellent agreement with open literature for limiting cases.

Divergence-free finite-difference method for 2D ideal MHD

The divergence-free finite-difference scheme for 2d ideal MHD using triangular unstructured staggered grids is described. In this approach the density, pressure and velocity fields are attributed to grid cells and magnetic field is defined by its normal components in cells edges. The HLLD approximate Riemann solver is used to compute the numerical flux of gas-dynamics variables at edges. It also provides the electrical field values used to compute the magnetic field. The magnetic field is interpolated into the cells using Raviart — Thomas basis functions. The algorythm is implemented in parallel numerical code. The method is tested on several well-known two-dimensional MHD problems.

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

Local Simulations of MRI turbulence with Meshless Methods

Abstract The magneto-rotational instability (MRI) is one of the most important processes in sufficiently ionized astrophysical disks. Grid-based simulations, especially those using the local shearing box approximation, provide a powerful tool to study the nonlinear turbulence the MRI produces. On the other hand, meshless methods have been widely used in cosmology, galactic dynamics, and planet formation, but have not been fully deployed on the MRI problem. We present local unstratified and vertically stratified MRI simulations with two meshless MHD schemes: a recent implementation of smoothed-particle magnetohydrodynamics (SPH MHD), and a meshless finite-mass (MFM) MHD scheme with constrained gradient divergence cleaning, as implemented in the GIZMO code. Concerning variants of the SPH hydro force formulation, we consider both the “vanilla” SPH and the PSPH variant included in GIZMO. We find, as expected, that the numerical noise inherent in these schemes significantly affects turbulence. Furthermore, a high-order kernel, free of the pairing instability, is necessary. Both schemes adequately simulate MRI turbulence in unstratified shearing boxes with net vertical flux. The turbulence, however, dies out in zero-net-flux unstratified boxes, probably due to excessive numerical dissipation. In zero-net-flux vertically stratified simulations, MFM can reproduce the MRI dynamo and its characteristic butterfly diagram for several tens of orbits before ultimately decaying. In contrast, extremely strong toroidal fields, as opposed to sustained turbulence, develop in equivalent simulations using SPH MHD. The latter unphysical state is likely caused by a combination of excessive artificial viscosity, numerical resistivity, and the relatively large residual errors in the divergence of the magnetic field.

Numerical study of MHD supersonic flow control

Supersonic MHD flow around a blunted body with a constant external magnetic field has been simulated for a number of geometries as well as a range of the flow parameters. Solvers based on Balbas-Tadmor MHD schemes and HLLC-Roe Godunov-type method have been developed within the OpenFOAM framework. The stability of the solution varies depending on the intensity of magnetic interaction The obtained solutions show the potential of MHD flow control and provide insights into for the development of the flow control system. The analysis of the results proves the applicability of numerical schemes, that are being used in the solvers. A number of ways to improve both the mathematical model of the process and the developed solvers are proposed.

Microphysics in the Gamma-Ray Burst Central Engine

Abstract We calculate the structure and evolution of a gamma-ray burst central engine where an accreting torus has formed around the newly born black hole. We study the general relativistic, MHD models and we self-consistently incorporate the nuclear equation of state. The latter accounts for the degeneracy of relativistic electrons, protons, and neutrons, and is used in the dynamical simulation, instead of a standard polytropic γ-law. The EOS provides the conditions for the nuclear pressure in the function of density and temperature, which evolve with time according to the conservative MHD scheme. We analyze the structure of the torus and outflowing winds, and compute the neutrino flux emitted through the nuclear reaction balance in the dense and hot matter. We also estimate the rate of transfer of the black-hole rotational energy to the bipolar jets. Finally, we elaborate on the nucleosynthesis of heavy elements in the accretion flow and the wind, through computations of the thermonuclear reaction network. We discuss the possible signatures of the radioactive element decay in the accretion flow. We suggest that further detailed modeling of the accretion flow in the GRB engine, together with its microphysics, may be a valuable tool to constrain the black-hole mass and spin. It can be complementary to the gravitational wave analysis if the waves are detected with an electromagnetic counterpart.

Splitting based scheme for three-dimensional MHD with dual time stepping

A new hybrid numerical scheme of combining an E-CUSP (Energy-Convective Upwind and Split Pressure) method for the fluid part and the Constrained Transport (CT) for the magnetic induction part is proposed. In order to avoid the occurrence of negative pressure in the reconstructed profiles and its updated value, a positivity preserving method is provided. Furthermore, the MHD equations are solved at each physical time step by advancing in pseudo time. The use of dual time stepping is beneficial in the computation since the use of dual time stepping allows the physical time step not to be limited by the corresponding values in the smallest cell and to be selected based on the numerical accuracy criterion. This newly established hybrid scheme combined with positivity preserving method and dual time technique has demonstrated the accurateness and robustness through numerical experiments of benchmark problems such as the 2D Orszag-Tang vortex problem and the 3D shock-cloud interaction problem.

VERTICAL STRUCTURE OF A SUPERNOVA-DRIVEN TURBULENT, MAGNETIZED INTERSTELLAR MEDIUM

Stellar feedback drives the circulation of matter from the disk to the halo of galaxies. We perform three-dimensional magnetohydrodynamic simulations of a vertical column of the interstellar medium with initial conditions typical of the solar circle in which supernovae drive turbulence and determine the vertical stratification of the medium. The simulations were run using a stable, positivity-preserving scheme for ideal MHD implemented in the FLASH code. We find that the majority (\approx 90 %) of the mass is contained in thermally-stable temperature regimes of cold molecular and atomic gas at T < 200 K or warm atomic and ionized gas at 5000 K < T < 10^{4.2} K, with strong peaks in probability distribution functions of temperature in both the cold and warm regimes. The 200 - 10^{4.2} K gas fills 50-60 % of the volume near the plane, with hotter gas associated with supernova remnants (30-40 %) and cold clouds (< 10 %) embedded within. At |z| ~ 1-2 kpc, transition-temperature (10^5 K) gas accounts for most of the mass and volume, while hot gas dominates at |z| > 3 kpc. The magnetic field in our models has no significant impact on the scale heights of gas in each temperature regime; the magnetic tension force is approximately equal to and opposite the magnetic pressure, so the addition of the field does not significantly affect the vertical support of the gas. The addition of a magnetic field does reduce the fraction of gas in the cold (< 200 K) regime with a corresponding increase in the fraction of warm (~ 10^4 K) gas. However, our models lack rotational shear and thus have no large-scale dynamo, which reduces the role of the field in the models compared to reality. The supernovae drive oscillations in the vertical distribution of halo gas, with the period of the oscillations ranging from ~ 30 Myr in the T < 200 K gas to ~ 100 Myr in the 10^6 K gas, in line with predictions by Walters & Cox.

A STABLE, ACCURATE METHODOLOGY FOR HIGH MACH NUMBER, STRONG MAGNETIC FIELD MHD TURBULENCE WITH ADAPTIVE MESH REFINEMENT: RESOLUTION AND REFINEMENT STUDIES

Performing a stable, long duration simulation of driven MHD turbulence with a high thermal Mach number and a strong initial magnetic field is a challenge to high-order Godunov ideal MHD schemes because of the difficulty in guaranteeing positivity of the density and pressure. We have implemented a robust combination of reconstruction schemes, Riemann solvers, limiters, and Constrained Transport EMF averaging schemes that can meet this challenge, and using this strategy, we have developed a new Adaptive Mesh Refinement (AMR) MHD module of the ORION2 code. We investigate the effects of AMR on several statistical properties of a turbulent ideal MHD system with a thermal Mach number of 10 and a plasma $\beta_0$ of 0.1 as initial conditions; our code is shown to be stable for simulations with higher Mach numbers ($M_rms = 17.3$) and smaller plasma beta ($\beta_0 = 0.0067$) as well. Our results show that the quality of the turbulence simulation is generally related to the volume-averaged refinement. Our AMR simulations show that the turbulent dissipation coefficient for supersonic MHD turbulence is about 0.5, in agreement with unigrid simulations.

PIERNIK MHD code – a multi–fluid, non–ideal extension of the relaxing–TVD scheme (III)

We present a new multi–fluid, grid MHD code PIERNIK, which is based on the Relaxing TVD scheme (Jin & Xin 1995). The original scheme (see Trac & Pen 2003; Pen et al 2003) has been extended by an addition of dynamically independent, but interacting fluids: dust and a diffusive cosmic ray gas, described within the fluid approximation, with an option to add other fluids in an easy way. The code has been equipped with shearing–box boundary conditions, and a selfgravity module, Ohmic resistivity module, as well as other facilities which are useful in astrophysical fluid–dynamical simulations. The code is parallelized by means of the MPI library. In this paper we present Ohmic resistivity extension of the original Relaxing TVD MHD scheme, and show examples of magnetic reconnection in cases of uniform and current–dependent resistivity prescriptions.

Turbulence in global simulations of magnetized thin accretion discs

We use a global magnetohydrodynamic simulation of a geometrically thin accretion disk to investigate the locality and detailed structure of turbulence driven by the magnetorotational instability (MRI). The model disk has an aspect ratio $H / R \simeq 0.07$, and is computed using a higher-order Godunov MHD scheme with accurate fluxes. We focus the analysis on late times after the system has lost direct memory of its initial magnetic flux state. The disk enters a saturated turbulent state in which the fastest growing modes of the MRI are well-resolved, with a relatively high efficiency of angular momentum transport $< < \alpha > > \approx 2.5 \times 10^{-2}$. The accretion stress peaks at the disk midplane, above and below which exists a moderately magnetized corona with patches of superthermal field. By analyzing the spatial and temporal correlations of the turbulent fields, we find that the spatial structure of the magnetic and kinetic energy is moderately well-localized (with correlation lengths along the major axis of $2.5H$ and $1.5H$ respectively), and generally consistent with that expected from homogenous incompressible turbulence. The density field, conversely, exhibits both a longer correlation length and a long correlation time, results which we ascribe to the importance of spiral density waves within the flow. Consistent with prior results, we show that the mean local stress displays a well-defined correlation with the local vertical flux, and that this relation is apparently causal (in the sense of the flux stimulating the stress) during portions of a global dynamo cycle. We argue that the observed flux-stress relation supports dynamo models in which the structure of coronal magnetic fields plays a central role in determining the dynamics of thin-disk accretion.

Astrophysical weighted particle magnetohydrodynamics

This paper presents applications of weighted meshless scheme for conservation laws to the Euler equations and the equations of ideal magnetohydrodynamics. The divergence constraint of the latter is maintained to the truncation error by a new meshless divergence cleaning procedure. The physics of the interaction between the particles is described by an one-dimensional Riemann problem in a moving frame. As a result, necessary diffusion which is required to treat dissipative processes is added automatically. As a result, our scheme has no free parameters that controls the physics of inter-particle interaction, with the exception of the number of the interacting neighbours which control the resolution and accuracy. The resulting equations have the form similar to SPH equations, and therefore existing SPH codes can be used to implement the weighed particle scheme. The scheme is validated in several hydrodynamic and MHD test cases. In particular, we demonstrate for the first time the ability of a meshless MHD scheme to model magneto-rotational instability in accretion disks.

RECONSTRUCTION OF THE CORONAL MAGNETIC FIELD USING THE CESE-MHD METHOD

We present a new implementation of the MHD relaxation method for reconstruction of the nearly force-free coronal magnetic field from a photospheric vector magnetogram. A new numerical MHD scheme is proposed to solve the full MHD equations by using the spacetime conservation-element and solution-element method. The bottom boundary condition is prescribed in a similar way as in the stress-and-relax method, by changing the transverse field incrementally to match the magnetogram, and other boundaries of the computational box are set by the nonreflecting boundary conditions. Applications to the well-known benchmarks for nonlinear force-free-field reconstruction, the Low & Lou force-free equilibria, validate the method and confirm its capability for future practical application, with observed magnetograms as inputs.

A Comparative Study of Divergence-Cleaning Techniques for Multi-Dimensional MHD Schemes

Several divergence-cleaning techniques for multi-dimensional Godunov-type magnetohydrodynamic schemes are comparatively investigated. We also propose a new divergence-cleaning technique that is improved from an earlier projection method to improve the robustness.

Relativistic magnetohydrodynamics in dynamical spacetimes: A new adaptive mesh refinement implementation

We have written and tested a new general relativistic magnetohydrodynamics (GRMHD) code, capable of evolving MHD fluids in dynamical spacetimes with adaptive-mesh refinement (AMR). Our code solves the Einstein-Maxwell-MHD system of coupled equations in full 3+1 dimensions, evolving the metric via the Baumgarte-Shapiro Shibata-Nakamura (BSSN) formalism and the MHD and magnetic induction equations via a conservative, high-resolution shock-capturing scheme. The induction equations are recast as an evolution equation for the magnetic vector potential, which exists on a grid that is staggered with respect to the hydrodynamic and metric variables. The divergenceless constraint div(B)=0 is enforced by the curl of the vector potential. Our MHD scheme is fully compatible with AMR, so that fluids at AMR refinement boundaries maintain div(B)=0. In simulations with uniform grid spacing, our MHD scheme is numerically equivalent to a commonly used, staggered-mesh constrained-transport scheme. We present code validation test results, both in Minkowski and curved spacetimes. They include magnetized shocks, nonlinear Alfv\'en waves, cylindrical explosions, cylindrical rotating disks, magnetized Bondi tests, and the collapse of a magnetized rotating star. Some of the more stringent tests involve black holes. We find good agreement between analytic and numerical solutions in these tests, and achieve convergence at the expected order.

Hamiltonian approach to hybrid plasma models

The Hamiltonian structures of several hybrid kinetic-fluid models are identified explicitly, upon considering collisionless Vlasov dynamics for the hot particles interacting with a bulk fluid. After presenting different pressure-coupling schemes for an ordinary fluid interacting with a hot gas, the paper extends the treatment to account for a fluid plasma interacting with an energetic ion species. Both current-coupling and pressure-coupling MHD schemes are treated extensively. In particular, pressure-coupling schemes are shown to require a transport-like term in the Vlasov kinetic equation, in order for the Hamiltonian structure to be preserved. The last part of the paper is devoted to studying the more general case of an energetic ion species interacting with a neutralizing electron background (hybrid Hall-MHD). Circulation laws and Casimir functionals are presented explicitly in each case.

New Laser Fusion and Its Gain by Intense Laser

The feasibility of a new approach of laser fusion in plasma without implosion has been proposed and is discussed using an intense laser. The cross section of the nuclear reaction is increased by enhancing the penetrability of nuclei through the Coulomb barrier. In this approach, an intense laser field of more than 100 PW was required to distort the Coulomb barrier to obtain enough penetrability. An energy gain even with Deuterium-Deuterium (D-D) reaction can be obtained using this scheme in Deuterium plasma. A reactor with neutron and direct conversion of charged particle beam individually is proposed. Charged particles from D-D reaction are guided to the end of the reactor and are directly converted by a MHD scheme into electric energy. The energy recovery rate is high and requires a small amount of laser energy, which may make the energy cost cheaper than that of a fission reactor.

High-order conservative finite difference GLM–MHD schemes for cell-centered MHD

We present and compare third- as well as fifth-order accurate finite difference schemes for the numerical solution of the compressible ideal MHD equations in multiple spatial dimensions. The selected methods lean on four different reconstruction techniques based on recently improved versions of the weighted essentially non-oscillatory (WENO) schemes, monotonicity preserving (MP) schemes as well as slope-limited polynomial reconstruction. The proposed numerical methods are highly accurate in smooth regions of the flow, avoid loss of accuracy in proximity of smooth extrema and provide sharp non-oscillatory transitions at discontinuities. We suggest a numerical formulation based on a cell-centered approach where all of the primary flow variables are discretized at the zone center. The divergence-free condition is enforced by augmenting the MHD equations with a generalized Lagrange multiplier yielding a mixed hyperbolic/parabolic correction, as in Dedner et al. [J. Comput. Phys. 175 (2002) 645–673]. The resulting family of schemes is robust, cost-effective and straightforward to implement. Compared to previous existing approaches, it completely avoids the CPU intensive workload associated with an elliptic divergence cleaning step and the additional complexities required by staggered mesh algorithms. Extensive numerical testing demonstrate the robustness and reliability of the proposed framework for computations involving both smooth and discontinuous features.