Constrained Transport Scheme Research Articles

A multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations

We present a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. This scheme relies on relaxation and splitting techniques and can be easily used at high order. A fully conservative version is not entropy satisfying but is observed experimentally to be more robust than standard constrained transport schemes at low plasma beta. At very low plasma beta and high Alfvén number, we have designed an entropy-satisfying version that is not conservative for the magnetic field but preserves admissible states and we switch locally a-priori between the two versions depending on the regime of plasma beta and Alfvén number. This strategy is robust in a wide range of standard MHD test cases, all performed at second order with a classic MUSCL-Hancock scheme.

Implementation of advanced Riemann solvers in a neutrino-radiation magnetohydrodynamics code in numerical relativity and its application to a binary neutron star merger

We implement advanced Riemann solvers HLLC and HLLD \cite{Mignone:2005ft,MUB:2009} together with an advanced constrained transport scheme \cite{Gardiner:2007nc} in a numerical-relativity neutrino-radiation magnetohydrodynamics code. We validate our implementation by performing a series of one- and multi-dimensional test problems for relativistic hydrodynamics and magnetohydrodynamics in both Minkowski spacetime and a static black hole spacetime. We find that the numerical solutions with the advanced Riemann solvers are more accurate than those with the HLLE solver \cite{DelZanna:2002rv}, which was originally implemented in our code. As an application to numerical relativity, we simulate an asymmetric binary neutron star merger leading to a short-lived massive neutron star both with and without magnetic fields. We find that the lifetime of the rotating massive neutron star formed after the merger and also the amount of the tidally-driven dynamical ejecta are overestimated when we employ the diffusive HLLE solver. We also find that the magnetorotational instability is less resolved when we employ the HLLE solver because of the solver's large numerical diffusivity. This causes a spurious enhancement both of magnetic winding resulting from large scale poloidal magnetic fields, and also of the energy of the outflow induced by magnetic pressure.

An Extension of Gmunu: General-relativistic Resistive Magnetohydrodynamics Based on Staggered-meshed Constrained Transport with Elliptic Cleaning

We present the implementation of general-relativistic resistive magnetohydrodynamics solvers and three divergence-free handling approaches adopted in the General-relativistic multigrid numerical (Gmunu) code. In particular, implicit–explicit Runge–Kutta schemes are used to deal with the stiff terms in the evolution equations for small resistivity. The three divergence-free handling methods are (i) hyperbolic divergence cleaning (also known as the generalized Lagrange multiplier), (ii) staggered-meshed constrained transport schemes, and (iii) elliptic cleaning through a multigrid solver, which is applicable in both cell-centered and face-centered (stagger grid) magnetic fields. The implementation has been tested with a number of numerical benchmarks from special-relativistic to general-relativistic cases. We demonstrate that our code can robustly recover from the ideal magnetohydrodynamics limit to a highly resistive limit. We also illustrate the applications in modeling magnetized neutron stars, and compare how different divergence-free handling methods affect the evolution of the stars. Furthermore, we show that the preservation of the divergence-free condition of the magnetic field when using staggered-meshed constrained transport schemes can be significantly improved by applying elliptic cleaning.

A Provably Positive, Divergence-free Constrained Transport Scheme for the Simulation of Solar Wind

Abstract In this paper, we present a provably positive, divergence-free constrained transport (CT) scheme to simulate the steady-state solar wind ambient with the three-dimensional magnetohydrodynamics numerical model. The positivity can be lost in two ways: one way is in the reconstruction process, and the other is in the updating process when the variables are advanced to the next time step. We adopt a self-adjusting strategy to bring the density and pressure into the permitted range in the reconstruction process, and use modified wave speeds in the Harten–Lax–van Leer flux to ensure the positivity in the updating process. The CT method can keep the magnetic fields divergence-free if the magnetic fields are divergence-free initially. Thus, we combine the least-squares reconstruction of the magnetic fields with the divergence-free constraints to make the magnetic fields globally solenoidal initially. Furthermore, we adopt a radial basis function method to interpolate variables at boundaries that can keep the magnetic field locally divergence-free. To verify the capability of the model in producing structured solar wind, the modeled results are compared with Parker Solar Probe (PSP) in situ observations during its first two encounters, as well as Wind observations at 1 au. Additionally, a solar maximum solar wind background is simulated to show the property of the model’s ability to preserve the positivity. The results show that the model can provide a relatively satisfactory comparison with PSP or Wind observations, and the divergence error is about 10−10 for all of the tests in this paper.

Systematic construction of upwind constrained transport schemes for MHD

The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or upwind, conservative codes. The evolution of the different magnetic field components, located at zone interfaces using a staggered representation, is achieved by calculating the electric field components at cell edges, in a way that has to be consistent with the Riemann solver used for the update of cell-centered fluid quantities at interfaces. Albeit several approaches have been undertaken, the purpose of this work is, on the one hand, to compare existing methods in terms of robustness and accuracy and, on the other, to extend the upwind constrained transport (UCT) method by Londrillo & Del Zanna (2004) [22] and Del Zanna et al. (2007) [23] for the systematic construction of new averaging schemes. In particular, we propose a general formula for the upwind fluxes of the induction equation which simply involves the information available from the base Riemann solver employed for the fluid part, provided it does not require full spectral decomposition, and 1D reconstructions of velocity and magnetic field components from nearby intercell faces to cell edges. Our results are presented here in the context of second-order schemes for classical MHD, but they can be easily generalized to higher than second order schemes, either based on finite volumes or finite differences, and to other physical systems retaining the same structure of the equations, such as that of relativistic or general relativistic MHD.

Impact of the Hall effect in star formation, improving the angular momentum conservation

We present here a minor modification of our numerical implementation of the Hall effect for the 2D Riemann solver used in Constrained Transport schemes, as described in a former paper. In the previous work, the tests showed that the angular momentum was not conserved during protostellar collapse simulations, with significant impact. By removing the whistler waves speed from the characteristic speeds of non-magnetic variables in the 1D Riemann solver, we were able to improve the angular momentum conservation in our test-case by one order of magnitude, while keeping the second-order numerical convergence of the scheme. We also reproduce the simulations of a previous study with consistent resistivities, the three non-ideal magnetohydrodynamic effects and initial rotation, and agree with their results. In this case, the violation of angular momentum conservation is negligible in regard to the total angular momentum and the angular momentum of the disk.

Impact of the Hall effect in star formation and the issue of angular momentum conservation

We present an implementation of the Hall term in the non-ideal magnetohydrodynamics equations into the adaptive-mesh-refinement codeRAMSESto study its impact on star formation. Recent works show that the Hall effect heavily influences the regulation of the angular momentum in collapsing dense cores, strengthening or weakening the magnetic braking. Our method consists of a modification of the two-dimensional constrained transport scheme. Our scheme shows convergence of second order in space and the frequency of the propagation of whistler waves is accurate. We confirm previous results, namely that during the collapse, the Hall effect generates a rotation of the fluid with a direction in the mid-plane that depends on the sign of the Hall resistivity, while counter-rotating envelopes develop on each side of the mid-plane. However, we find that the predictability of our numerical results is severely limited. The angular momentum is not conserved in any of our dense core-collapse simulations with the Hall effect: a large amount of angular momentum is generated within the first Larson core, a few hundred years after its formation, without compensation by the surrounding gas. This issue is not mentioned in previous studies and may be correlated to the formation of the accretion shock on the Larson core. We expect that this numerical effect could be a serious issue in star formation simulations.

Non-ideal magnetohydrodynamics on a moving mesh

In certain astrophysical systems the commonly employed ideal magnetohydrodynamics (MHD) approximation breaks down. Here, we introduce novel explicit and implicit numerical schemes of ohmic resistivity terms in the moving-mesh code AREPO. We include these non-ideal terms for two MHD techniques: the Powell 8-wave formalism and a constrained transport scheme, which evolves the cell-centred magnetic vector potential. We test our implementation against problems of increasing complexity, such as one- and two-dimensional diffusion problems, and the evolution of progressive and stationary Alfv\'en waves. On these test problems, our implementation recovers the analytic solutions to second-order accuracy. As first applications, we investigate the tearing instability in magnetized plasmas and the gravitational collapse of a rotating magnetized gas cloud. In both systems, resistivity plays a key role. In the former case, it allows for the development of the tearing instability through reconnection of the magnetic field lines. In the latter, the adopted (constant) value of ohmic resistivity has an impact on both the gas distribution around the emerging protostar and the mass loading of magnetically driven outflows. Our new non-ideal MHD implementation opens up the possibility to study magneto-hydrodynamical systems on a moving mesh beyond the ideal MHD approximation.

DISCO: A 3D MOVING-MESH MAGNETOHYDRODYNAMICS CODE DESIGNED FOR THE STUDY OF ASTROPHYSICAL DISKS

ABSTRACT This work presents the publicly available moving-mesh magnetohydrodynamics (MHD) code DISCO. DISCO is efficient and accurate at evolving orbital fluid motion in two and three dimensions, especially at high Mach numbers. DISCO employs a moving-mesh approach utilizing a dynamic cylindrical mesh that can shear azimuthally to follow the orbital motion of the gas. The moving mesh removes diffusive advection errors and allows for longer time-steps than a static grid. MHD is implemented in DISCO using an HLLD Riemann solver and a novel constrained transport (CT) scheme that is compatible with the mesh motion. DISCO is tested against a wide variety of problems, which are designed to test its stability, accuracy, and scalability. In addition, several MHD tests are performed which demonstrate the accuracy and stability of the new CT approach, including two tests of the magneto-rotational instability, one testing the linear growth rate and the other following the instability into the fully turbulent regime.

A moving mesh unstaggered constrained transport scheme for magnetohydrodynamics

We present a constrained transport (CT) algorithm for solving the 3D ideal magnetohydrodynamic (MHD) equations on a moving mesh, which maintains the divergence-free condition on the magnetic field to machine-precision. Our CT scheme uses an unstructured representation of the magnetic vector potential, making the numerical method simple and computationally efficient. The scheme is implemented in the moving mesh code Arepo. We demonstrate the performance of the approach with simulations of driven MHD turbulence, a magnetized disc galaxy, and a cosmological volume with primordial magnetic field. We compare the outcomes of these experiments to those obtained with a previously implemented Powell divergence-cleaning scheme. While CT and the Powell technique yield similar results in idealized test problems, some differences are seen in situations more representative of astrophysical flows. In the turbulence simulations, the Powell cleaning scheme artificially grows the mean magnetic field, while CT maintains this conserved quantity of ideal MHD. In the disc simulation, CT gives slower magnetic field growth rate and saturates to equipartition between the turbulent kinetic energy and magnetic energy, whereas Powell cleaning produces a dynamically dominant magnetic field. Such difference has been observed in adaptive-mesh refinement codes with CT and smoothed-particle hydrodynamics codes with divergence-cleaning. In the cosmological simulation, both approaches give similar magnetic amplification, but Powell exhibits more cell-level noise. CT methods in general are more accurate than divergence-cleaning techniques, and, when coupled to a moving mesh can exploit the advantages of automatic spatial/temporal adaptivity and reduced advection errors, allowing for improved astrophysical MHD simulations.

Stability analysis and error estimates of an exactly divergence-free method for the magnetic induction equations

In this paper, we consider an exactly divergence-free scheme to solve the magnetic induction equations. This problem is motivated by the numerical simulations of ideal magnetohydrodynamic (MHD) equations, a nonlinear hyperbolic system with a divergence-free condition on the magnetic field. Computational methods without satisfying such condition may lead to numerical instability. One class of methods, constrained transport schemes, is widely used as divergence-free treatments. So far there is not much analysis available for such schemes. In this work, we take an exactly divergence-free scheme proposed by [Li and Xu, J. Comput. Phys. 231 (2012) 2655–2675] as a candidate of the constrained transport schemes, and adapt it to solve the magnetic induction equations. For the resulting scheme applied to the equations with a constant velocity field, we carry out von Neumann analysis for numerical stability on uniform meshes. We also establish the stability and error estimates based on energy methods. In particular, we identify the stability mechanism due to the spatial and temporal discretizations, and the role of the exactly divergence-free property of the numerical solution for stability. The analysis based on energy methods can be extended to non-uniform meshes, and they can also be applied to the magnetic induction equations with a variable velocity field, which is more relevant to the MHD simulations.

A constrained transport scheme for MHD on unstructured static and moving meshes

Magnetic fields play an important role in many astrophysical systems and a detailed understanding of their impact on the gas dynamics requires robust numerical simulations. Here we present a new method to evolve the ideal magnetohydrodynamic (MHD) equations on unstructured static and moving meshes that preserves the magnetic field divergence-free constraint to machine precision. The method overcomes the major problems of using a cleaning scheme on the magnetic fields instead, which is non-conservative, not fully Galilean invariant, does not eliminate divergence errors completely, and may produce incorrect jumps across shocks. Our new method is a generalization of the constrained transport (CT) algorithm used to enforce the $\nabla\cdot \mathbf{B}=0$ condition on fixed Cartesian grids. Preserving $\nabla\cdot \mathbf{B}=0$ at the discretized level is necessary to maintain the orthogonality between the Lorentz force and $\mathbf{B}$. The possibility of performing CT on a moving mesh provides several advantages over static mesh methods due to the quasi-Lagrangian nature of the former (i.e., the mesh generating points move with the flow), such as making the simulation automatically adaptive and significantly reducing advection errors. Our method preserves magnetic fields and fluid quantities in pure advection exactly.

FISH: A THREE-DIMENSIONAL PARALLEL MAGNETOHYDRODYNAMICS CODE FOR ASTROPHYSICAL APPLICATIONS

FISH is a fast and simple ideal magneto-hydrodynamics code that scales to ~10 000 processes for a Cartesian computational domain of ~1000^3 cells. The simplicity of FISH has been achieved by the rigorous application of the operator splitting technique, while second order accuracy is maintained by the symmetric ordering of the operators. Between directional sweeps, the three-dimensional data is rotated in memory so that the sweep is always performed in a cache-efficient way along the direction of contiguous memory. Hence, the code only requires a one-dimensional description of the conservation equations to be solved. This approach also enable an elegant novel parallelisation of the code that is based on persistent communications with MPI for cubic domain decomposition on machines with distributed memory. This scheme is then combined with an additional OpenMP parallelisation of different sweeps that can take advantage of clusters of shared memory. We document the detailed implementation of a second order TVD advection scheme based on flux reconstruction. The magnetic fields are evolved by a constrained transport scheme. We show that the subtraction of a simple estimate of the hydrostatic gradient from the total gradients can significantly reduce the dissipation of the advection scheme in simulations of gravitationally bound hydrostatic objects. Through its simplicity and efficiency, FISH is as well-suited for hydrodynamics classes as for large-scale astrophysical simulations on high-performance computer clusters. In preparation for the release of a public version, we demonstrate the performance of FISH in a suite of astrophysically orientated test cases.

AN EXPLICIT SCHEME FOR INCORPORATING AMBIPOLAR DIFFUSION IN A MAGNETOHYDRODYNAMICS CODE

We describe a method for incorporating ambipolar diffusion in the strong coupling approximation into a multidimensional magnetohydrodynamic (MHD) code based on the total variation diminishing scheme. Contributions from ambipolar diffusion terms are included by explicit finite-difference operators in a fully unsplit way, maintaining second-order accuracy. The divergence-free condition of magnetic fields is exactly ensured at all times by a flux-interpolated constrained transport scheme. The super time-stepping method is used to accelerate the time step in high-resolution calculations and/or in strong ambipolar diffusion. We perform two test problems, the steady-state oblique C-type shocks and the decay of Alfven waves, confirming the accuracy and robustness of our numerical approach. Results from the simulations of the compressible MHD turbulence with ambipolar diffusion show the flexibility of our method as well as its ability to follow complex MHD flows in the presence of ambipolar diffusion. These simulations show that the dissipation rate of the MHD turbulence is strongly affected by the strength of ambipolar diffusion.

High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method

This paper extends the central finite-volume schemes of Liu et al. [Y. Liu, C.-W. Shu, E. Tadmor, M. Zhang, Non-oscillatory hierarchical reconstruction for central and finite-volume schemes, Commun. Comput. Phys. 2 (2007) 933–963] on overlapping cells to the magneto-hydrodynamic (MHD) equations. In particular, we propose a high order divergence-free reconstruction for the magnetic field that uses the face-centered values. We also advance the magnetic field with a high order constrained transport (CT) scheme to preserve the divergence-free condition to machine round-off error. The overlapping cells are natural to be used to calculate the electric field flux without an averaging procedure. We have developed a third-order scheme which is verified by the numerical experiments. Other higher order schemes can be constructed accordingly. Our central constrained transport schemes do not need characteristic decomposition, and are easy to code and combine with un-split discretization of the source and parabolic terms. The overlapping cell representation of the solution is also used to develop more compact reconstruction and less dissipative schemes. The high resolution is achieved by non-oscillatory hierarchical reconstruction, which does not require characteristic decomposition either. The numerical comparisons show that the central schemes with non-CT perform as well as with CT for most of problems. Numerical examples are given to demonstrate efficacy of the new schemes.

Divergence- and Curl-Preserving Prolongation and Restriction Formulas

We present new second-order prolongation and restriction formulas which preserve the divergence and, in some cases, the curl of a discretized vector field. The formulas are suitable for adaptive and hierarchical mesh algorithms with a factor-of-2 linear resolution change. We examine both staggered and collocated discretizations for the vector field on two- and three-dimensional Cartesian grids. The new formulas can be used in combination with numerical schemes that require a divergence-free solution in some discrete sense, such as the constrained transport schemes of computational magnetohydrodynamics. We also obtain divergence-preserving interpolation functions which may be used for streamline or field line tracing.

The ∇·B=0 Constraint in Shock-Capturing Magnetohydrodynamics Codes

Seven schemes to maintain the ∇·B=0 constraint numerically are compared. All these algorithms can be combined with shock-capturing Godunov type base schemes. They fall into three categories: the eight-wave formulation maintains the constraint to truncation error, the projection scheme enforces the constraint in some discretization by projecting the magnetic field, while the five different versions of the constrained transport/central difference type schemes conserve ∇·B to machine accuracy in some discretization for every grid cell. It is shown that the three constrained transport algorithms, which have been introduced recently, can be recast into pure finite volume schemes, and the staggered representation of the magnetic field is unnecessary. Another two new and simple central difference based algorithms are introduced. The properties of the projection scheme are discussed in some detail, and I prove that it has the same order of accuracy as the base scheme even for discontinuous solutions. I describe a flexible and efficient implementation of the projection scheme using conjugate gradient type iterative methods. Generalizations to resistive MHD, to axial symmetry, and to non-Cartesian grids are given for all schemes. The theoretical discussion is followed by numerical tests, where the robustness, accuracy, and efficiency of the seven schemes and the base scheme can be directly compared. All simulations are done with the Versatile Advection Code, in which several shock-capturing base schemes are implemented. Although the eight-wave formulation usually works correctly, one of the numerical tests demonstrates that its non-conservative nature can occasionally produce incorrect jumps across strong discontinuities. Based on a large number of tests, the projection scheme, one of the new central difference based schemes, and one of the constrained transport schemes are found to be the most accurate and reliable among the examined methods.