Balsara (2001, J. Comput. Phys., 174, 614) showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD was shown in Balsara (2004, ApJS, 151, 149). Second order accurate divergence-free schemes for MHD have shown themselves to be very useful in several areas of science and engineering. However, certain computational MHD problems would be much benefited if the schemes had third and higher orders of accuracy. In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free weighted essentially non-oscillatory (WENO) schemes for MHD that have order of accuracy better than second. A multistage Runge-Kutta time integration is used to ensure that the temporal accuracy matches the spatial accuracy. Accuracy analysis is carried out and it is shown that the schemes meet their design accuracy for smooth problems. Stringent tests are also presented showing that the schemes perform well on those tests.
The present paper introduces a class of finite volume schemes of increasing order of accuracy in space and time for hyperbolic systems that are in conservation form. This paper specifically focuses on Euler system that is used for modeling the flow of neutral fluids and the divergence-free, ideal magnetohydrodynamics (MHD) system that is used for large scale modeling of ionized plasmas. Efficient techniques for weighted essentially non-oscillatory (WENO) interpolation have been developed for finite volume reconstruction on structured meshes. We also present a new formulation of the ADER (for A…
In this paper, we introduce a high-order accurate constrained transport type finite volume method to solve ideal magnetohydrodynamic equations on two-dimensional triangular meshes. A new divergence-free WENO-based reconstruction method is developed to maintain exactly divergence-free evolution of the numerical magnetic field. A new weighted flux interpolation approach is also developed to compute the z-component of the electric field at vertices of grid cells. We also present numerical examples to demonstrate the accuracy and robustness of the proposed scheme.
In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton-Jacobi (H-J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme [10] for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme …
Due to increase in computing power, high-order Eulerian schemes will likely become instrumental for the simulations of turbulence and magnetic field amplification in astrophysical fluids in the next years. We present the implementation of a fifth order weighted essentially non-oscillatory scheme for constrained-transport magnetohydrodynamics into the code WOMBAT. We establish the correctness of our implementation with an extensive number tests. We find that the fifth order scheme performs as accurately as a common second order scheme at half the resolution. We argue that for a given solution q…
The relativistic magnetohydrodynamics (RMHD) set of equations has recently seen increased use in astrophysical computations. Even so, RMHD codes remain fragile. The reconstruction can sometimes yield superluminal velocities in certain parts of the mesh. In this paper we present a reconstruction strategy that overcomes this problem by making a single conservative to primitive transformation per cell followed by higher order WENO reconstruction on a carefully chosen set of primitives that guarantee subluminal reconstruction of the flow variables. For temporal evolution via a predictor step we al…
Balsara (2001) showed the importance of divergence-free reconstruction in adaptive mesh refinement problems for magnetohydrodynamics (MHD) and the importance of the same for designing robust second order schemes for MHD was shown in Balsara (2004). In this paper we show that the reconstruction of divergence-free vector fields can be carried out with better than second order accuracy. As a result, we design divergence-free RKDG and hybrid RKDG+HWENO schemes for MHD that have accuracies that are better than second order. Accuracy analysis is carried out and it is shown that the schemes meet thei…
We present a new method for evolving the equations of magnetohydrodynamics (both Newtonian and relativistic) that is capable of maintaining a divergence-free magnetic field ($\nabla \cdot \mathbf{B} = 0$) on adaptively refined, conformally moving meshes. The method relies on evolving the magnetic vector potential and then using it to reconstruct the magnetic fields. The advantage of this approach is that the vector potential is not subject to a constraint equation in the same way the magnetic field is, and so can be refined and moved in a straightforward way. We test this new method against a …
We present the first high order one-step ADER-WENO finite volume scheme with Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial accuracy is obtained through a WENO reconstruction, while a high order one-step time discretization is achieved using a local space-time discontinuous Galerkin predictor method. Due to the one-step nature of the underlying scheme, the resulting algorithm is particularly well suited for an AMR strategy on space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR property has been implemented 'cell-by-cell', with a stan…
Finite difference method was extended to unstructured meshes to solve Euler equations. The spatial discretization is made of two steps. First, numerical fluxes are computed at the middle point of each edge with high order accuracy. In this step, a one-dimensional curvilinear stencil is assembled for each edge to perform one-dimensional fast non-uniform WENO interpolation derived in this paper, which is much easier and faster than multi-dimensional interpolation. The second step is to compute the divergence of fluxes at each vertex from the fluxes at nearby edges and vertices by least square ap…
Modified-gradient methods for exact divergence-free in meshless magnetohydrodynamics
Hubei Normal University（湖北师范大学）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）；University of Chinese Academy of Sciences（中国科学院大学）；Beijing Normal University（北京师范大学）；Zhengzhou University（郑州大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
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. Three divergence-free handling methods are (i) hyperbolic divergence cleaning through a generalised Lagrange multiplier (GLM); (ii) staggered-meshed constrained transport (CT) schemes and (iii) elliptic cleaning though multigrid (MG) solver which is …
Systematic Comparison between Constrained Transport and Mixed Divergence Cleaning Methods for Astrophysical Magnetohydrodynamic Simulations
Astronomical Institute, Tohoku University（东北大学天文学研究所）；École Normale Supérieure de Lyon, Centre de Recherche Astrophysique de Lyon（里昂高等师范学校，里昂天体物理研究中心）；Humanities and Sciences/Museum Careers, Musashino Art University（武藏野美术大学人文科学与博物馆职业）；Center for Computational Astrophysics, National Astronomical Observatory of Japan（日本国立天文台计算天体物理学中心）
We present a constrained formulation of Dedner et al's hyperbolic/parabolic divergence cleaning scheme for enforcing the \nabla\dot B = 0 constraint in Smoothed Particle Magnetohydrodynamics (SPMHD) simulations. The constraint we impose is that energy removed must either be conserved or dissipated, such that the scheme is guaranteed to decrease the overall magnetic energy. This is shown to require use of conjugate numerical operators for evaluating \nabla\dot B and \nablaψ in the SPMHD cleaning equations. The resulting scheme is shown to be stable at density jumps and free boundaries, in contr…
A description is given of the algorithms implemented in the AstroBEAR adaptive mesh refinement code for ideal magnetohydrodynamics. The code provides several high resolution, shock capturing schemes which are constructed to maintain conserved quantities of the flow in a finite volume sense. Divergence free magnetic field topologies are maintained to machine precision by collating the components of the magnetic field on a cell-interface staggered grid and utilizing the constrained transport approach for integrating the induction equations. The maintenance of magnetic field topologies on adaptiv…
In this work we present a general strategy for constructing multidimensional Riemann solvers with a single intermediate state, with particular attention paid to detailing the two-dimensional Riemann solver. This is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressur…
A high order finite difference method is proposed for unstructured meshes to simulate compressible inviscid/viscous flows with/without discontinuities. In this method, based on the strong form equation, the divergence of the flux on each vertex is computed directly from fluxes nearby by means of high order least-square. In order to capture discontinuities, numerical flux of high order accuracy is calculated on each edge and serves as supporting data of the least-square computation of the divergence. The high accuracy of the numerical flux depends on the high order WENO interpolation on each ed…
Modern astrophysical simulations aim to accurately model an ever-growing array of physical processes, including the interaction of fluids with magnetic fields, under increasingly stringent performance and scalability requirements driven by present-day trends in computing architectures. Discontinuous Galerkin methods have recently gained some traction in astrophysics, because of their arbitrarily high order and controllable numerical diffusion, combined with attractive characteristics for high performance computing. In this paper, we describe and test our implementation of a discontinuous Galer…
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 present a virtual element method (VEM) for the numerical approximation of the electromagnetics subsystem of the resistive magnetohydrodynamics (MHD) model in two spatial dimensions. The major advantages of the virtual element method include great flexibility of polygonal meshes and automatic divergence-free constraint on the magnetic flux field. In this work, we rigorously prove the well-posedness of the method and the solenoidal nature of the discrete magnetic flux field. We also derive stability energy estimates. The design of the method includes three choices for the construction of the …
There is a great need in several areas of astrophysics and space-physics to carry out high order of accuracy, divergence-free MHD simulations on spherical meshes. This requires us to pay careful attention to the interplay between mesh quality and numerical algorithms. Methods have been designed that fundamentally integrate high order isoparametric mappings with the other high accuracy algorithms that are needed for divergence-free MHD simulations on geodesic meshes. The goal of this paper is to document such algorithms that are implemented in the geodesic mesh version of the RIEMANN code. The …
Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision of the adaptive computations, while reducing CPU time and memory requirements. Implementation issues of the open source code CARMEN-MHD are discussed. To illustrate its precision and efficiency…
Force-free plasmas are a good approximation where the plasma pressure is tiny compared with the magnetic pressure, which is the case during the cold vertical displacement event (VDE) of a major disruption in a tokamak. On time scales long compared with the transit time of Alfven waves, the evolution of a force-free plasma is most efficiently described by the quasi-static magnetohydrodynamic (MHD) model, which ignores the plasma inertia. Here we consider a regularized quasi-static MHD model for force-free plasmas in tokamak disruptions and propose a mimetic finite difference (MFD) algorithm. Th…
GAMER, a parallel Graphic-processing-unit-accelerated Adaptive-MEsh-Refinement hydrodynamic code, has been extended to support magnetohydrodynamics (MHD) with both the corner-transport-upwind (CTU) and MUSCL-Hancock schemes and the constraint transport (CT) technique. The divergent preserving operator for adaptive mesh refinement (AMR) has been applied to reinforce the divergence-free constraint on the magnetic field. GAMER-MHD has fully exploited the concurrent executions between the GPU MHD solver and other CPU computation pertinent to AMR. We perform various standard tests to demonstrate th…
An interpolation method to evaluate magnetic fields given unstructured, scattered magnetic data is presented. The method is based on the reconstruction of the global magnetic field using a superposition of orthogonal functions. The coefficients of the expansion are obtained by minimizing a cost function defined as the L^2 norm of the difference between the ground truth and the reconstructed magnetic field evaluated on the training data. The divergence-free condition is incorporated as a constrain in the cost function allowing the method to achieve arbitrarily small errors in the magnetic field…
Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
A new multidimensional simulation code for relativistic two-fluid electrodynamics (RTFED) is described. The basic equations consist of the full set of Maxwell's equations coupled with relativistic hydrodynamic equations for separate two charged fluids, representing the dynamics of either an electron-positron or an electron-proton plasma. It can be recognized as an extension of conventional relativistic magnetohydrodynamics (RMHD). Finite resistivity may be introduced as a friction between the two species, which reduces to resistive RMHD in the long wavelength limit without suffering from a sin…
Based on a 14-moment closure for non-resistive (general-) relativistic viscous plasmas, we describe a new numerical scheme that is able to handle all first-order dissipative effects (heat conduction, bulk and shear viscosities), as well the anisotropies induced by the presence of magnetic fields. The latter is parameterized in terms of a thermal gyrofrequency or, equivalently, a thermal Larmor radius and allows to correctly capture the thermal Hall effect. By solving an extended Israel-Stewart-like system for the dissipative quantities that enforces algebraic constraints via stiff-relaxation, …
We present a new magnetohydrodynamic (MHD) simulation code with the aim of providing accurate numerical solutions to astrophysical phenomena where discontinuities, shock waves, and turbulence are inherently important. The code implements the HLLD approximate Riemann solver, the fifth-order-monotonicity-preserving interpolation (MP5) scheme, and the hyperbolic divergence cleaning method for a magnetic field. This choice of schemes significantly improved numerical accuracy and stability, and saved computational costs in multidimensional problems. Numerical tests of one- and two-dimensional probl…
We present the implementation of a three-dimensional, second order accurate Godunov-type algorithm for magneto-hydrodynamic (MHD), in the adaptive-mesh-refinement (AMR) cosmological code {\tt CHARM}. The algorithm is based on the full 12-solve spatially unsplit Corner-Transport-Upwind (CTU) scheme. The fluid quantities are cell-centered and are updated using the Piecewise-Parabolic-Method (PPM), while the magnetic field variables are face-centered and are evolved through application of the Stokes theorem on cell edges via a Constrained-Transport (CT) method. The multidimensional MHD source ter…
The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In the first part of this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretisation of the $\mathbf{B}$-$\mathbf{E}$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling n…
We present a new code designed to solve the equations of classical ideal magneto-hydrodynamics (MHD) in three dimensions, submitted to a constant gravitational field. The purpose of the code centers on the analysis of solar phenomena within the photosphere-corona region. We present 1D and 2D standard tests to demonstrate the quality of the numerical results obtained with our code. As solar tests we present the transverse oscillations of Alfvenic pulses in coronal loops using a 2.5D model, and as 3D tests we present the propagation of impulsively generated MHD-gravity waves and vortices in the …
