Two-dimensional Ideal Magnetohydrodynamics Research Articles

A robust and contact resolving Riemann solver for the two-dimensional ideal magnetohydrodynamics equations

This paper presents a new cell-centered numerical method for ideal magnetohydrodynamics (MHD) that can be used in Lagrangian or Eulerian discretization. A two-dimensional Riemann solver based on the HLLC-type MHD method is established, which can be viewed as an extension from the HLLC-2D in hydrodynamics. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. It transforms naturally from Lagrangian setting to the Eulerian setting in terms of grid nodal velocity, and gains benefits of the Lagrangian nature of the scheme. In the Lagrangian approach, the finite volume scheme itself can keep the magnetic field divergence-free strictly, while in the Eulerian case, a special constrained transport (CT) algorithm is constructed from the discontinuous fluxes on cell interfaces to ensure solenoidal nature again. Numerical tests are presented to demonstrate the performance of this new solver and compare the difference between the Lagrangian and Eulerian methods.

A Robust and Contact Resolving Riemann Solver for the Two-Dimensional Ideal Magnetohydrodynamics Equations

A Robust and Contact Resolving Riemann Solver for the Two-Dimensional Ideal Magnetohydrodynamics Equations

Gas-Kinetic BGK Scheme for Three Dimensional Magnetohydrodynamics

The gas-kinetic theory based flux splitting method has been successfully proposed for solving oneand two-dimensional ideal magnetohydrodynamics by Xu et al. [J. Comput. Phys., 1999; 2000], respectively. This paper extends the kinetic method to solve three-dimensional ideal magnetohydrodynamics equations, where an adaptive parameter η is used to control the numerical dissipation in the flux splitting method. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy and resolve strong discontinuous waves in three dimensional ideal MHD problems. AMS subject classifications: 76M25, 76W05

Numerical Simulations of Slow Standing Waves in a Curved Solar Coronal Loop

We consider slow standing waves that are impulsively excited in a curved solar coronal loop. The numerical model we implement includes the effect of nonlinearity in the frame of two-dimensional ideal magnetohydrodynamics. We discuss the role of curved magnetic field lines and of the slow and fast pulses overlapping at one of the loop's footpoints in the excitation and attenuation of slow standing waves. We find that slow waves can be excited faster in curved loops than in slabs due to the combined effect of the pulse inside and outside the loop.

An adaptive moving mesh method for two-dimensional ideal magnetohydrodynamics

This paper presents an adaptive moving mesh algorithm for two-dimensional (2D) ideal magnetohydrodynamics (MHD) that utilizes a staggered constrained transport technique to keep the magnetic field divergence-free. The algorithm consists of two independent parts: MHD evolution and mesh-redistribution. The first part is a high-resolution, divergence-free, shock-capturing scheme on a fixed quadrangular mesh, while the second part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative-interpolation formula is used to calculate the remapped cell-averages of the mass, momentum, and total energy on the resulting new mesh; the magnetic potential is remapped to the new mesh in a non-conservative way and is reconstructed to give a divergence-free magnetic field on the new mesh. Several numerical examples are given to demonstrate that the proposed method can achieve high numerical accuracy, track and resolve strong shock waves in ideal MHD problems, and preserve divergence-free property of the magnetic field. Numerical examples include the smooth Alfvén wave problem, 2D and 2.5D shock tube problems, two rotor problems, the stringent blast problem, and the cloud–shock interaction problem.

Linear and nonlinear stability analysis for two-dimensional ideal magnetohydrodynamics with incompressible flows

The equilibrium and Lyapunov stability properties for two-dimensional ideal magnetohydrodynamic (MHD) plasmas with incompressible and homogeneous (i.e., constant density) flows are investigated. In the unperturbed steady state, both the velocity and magnetic field are nonzero and have three components in a Cartesian coordinate system with translational symmetry (i.e., one ignorable spatial coordinate). It is proved that (a) the solutions of the ideal MHD steady state equations with incompressible and homogeneous flows in the plane are also valid for equilibria with the axial velocity component being a free flux function and the axial magnetic field component being a constant, (b) the conditions of linearized Lyapunov stability for these MHD flows in the planar case (in which the fields have only two components) are also valid for symmetric equilibria that have a nonplanar velocity field component as well as a nonplanar magnetic field component. On using the method of convexity estimates, nonlinear stability conditions are established.

A CWENO-type central-upwind scheme for ideal MHD equations

A simple fourth-order, central-upwind scheme based on central weighted essentially non-oscillatory (CWENO) reconstruction is proposed in this paper for computing the approximate solutions of one- and two-dimensional ideal magnetohydrodynamics (MHD) equations with high-resolution. Since the non-uniform width of the different local Riemann fans is calculated more accurately, the central-upwind schemes enjoy a much smaller numerical viscosity as well as the staggering between two neighboring sets of grids is avoided. Due to the central-upwind scheme is combined with the fourth-order CWENO reconstruction, the scheme we present has the non-oscillatory behavior. The numerical results show the desired accuracy, high-resolution, and robustness of our method.

On the behavior of hyperbolic neutral points in two-dimensional ideal magnetohydrodynamics.

We study ideal incompressible magnetohydrodynamics in two dimensions. We obtain an exponential estimate on the closing of the angle at hyperbolic saddle points of the magnetic stream function under the assumption that the velocity remains bounded. The analytic results are supported by numerical simulations. These results give evidence against a standard scenario for singularity formation for these equations.

On current sheets in two-dimensional ideal magnetohydrodynamics caused by pressure perturbations

In this paper it is shown that in ideal magnetohydrodynamics (IMHD) a two-dimensional equilibrium with an X-shaped neutral point (X-point) configuration reacts to small and smooth perturbations of the plasma pressure by the development of sheets with singular current density while the X-point structure is replaced by a different kind of configuration containing either cusp-like or T-shaped singularities. This behavior is demonstrated by computing the time dependent response to a suitable initial perturbation using a relaxation method within ideal MHD. The two-dimensional configurations considered in this paper are motivated by models of space plasma systems like, e.g. the Earth’s magnetotail containing a distant X-point where perturbations caused by the solar wind may force the formation of current sheets and the subsequent occurrence of magnetic reconnection.

Two-dimensional ideal magnetohydrodynamics and differential geometry

It is shown that equations of two-dimensional ideal magnetohydrodynamics may be regarded as geodesic equations on appropriate infinite dimensional Lie group. The physical interpretation of such a geometric picture is given using an analogy with two-and-a-half-dimensional ideal hydrodynamics. The sectional curvature responsible for the separation of neighbouring geodesics and, hence, for stability is calculated.