A New Implementation of the Magnetohydrodynamics-Relaxation Method for Nonlinear Force-Free Field Extrapolation in the Solar Corona

Magnetic field in the solar corona is usually extrapolated from photospheric vector magnetogram using a nonlinear force-free field (NLFFF) model. NLFFF extrapolation needs a considerable effort to be devoted for its numerical realization. In this paper we present a new implementation of the magnetohydrodynamics (MHD)-relaxation method for NLFFF extrapolation. The magneto-frictional approach which is introduced for speeding the relaxation of the MHD system is novelly realized by the spacetime conservation-element and solution-element (CESE) scheme. A magnetic field splitting method is used to further improve the computational accuracy. The bottom boundary condition is prescribed by changing the transverse field incrementally to match the magnetogram, and all other artificial boundaries of the computational box are simply fixed. We examine the code by two types of NLFFF benchmark tests, the Low & Lou (1990) semi-analytic force-free solutions and a more realistic solar-like case constructed by van Ballegooijen et al. (2007). The results show that our implementation are successful and versatile for extrapolations of either the relatively simple cases or the rather complex cases which need significant rebuilding of the magnetic topology, e.g., a flux rope. We also compute a suite of metrics to quantitatively analyze the results and demonstrate that the performance of our code in extrapolation accuracy basically reaches the same level of the present best-performing code, e.g., that developed by Wiegelmann (2004).Comment: Accept by ApJ, 45 pages, 13 figure

MHD Simulation of the Inner-Heliospheric Magnetic Field

Maps of the radial magnetic field at a heliocentric distance of ten solar radii are used as boundary conditions in the MHD code CRONOS to simulate a 3D inner-heliospheric solar wind emanating from the rotating Sun out to 1 AU. The input data for the magnetic field are the result of solar surface flux transport modelling using observational data of sunspot groups coupled with a current sheet source surface model. Amongst several advancements, this allows for higher angular resolution than that of comparable observational data from synoptic magnetograms. The required initial conditions for the other MHD quantities are obtained following an empirical approach using an inverse relation between flux tube expansion and radial solar wind speed. The computations are performed for representative solar minimum and maximum conditions, and the corresponding state of the solar wind up to the Earths orbit is obtained. After a successful comparison of the latter with observational data, they can be used to drive outer-heliospheric models.Comment: for associated wmv movie files accompanying Figure 7, see http://www.tp4.rub.de/~tow/max.wmv and http://www.tp4.rub.de/~tow/min.wm

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier-Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates $(\xi, \eta, \zeta)$. The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge-Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., $\mathbf{B}=\textbf{0}$) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme

Three-Dimensional Navier-Stokes Calculations Using the Modified Space-Time CESE Method

The space-time conservation element solution element (CESE) method is modified to address the robustness issues of high-aspect-ratio, viscous, near-wall meshes. In this new approach, the dependent variable gradients are evaluated using element edges and the corresponding neighboring solution elements while keeping the original flux integration procedure intact. As such, the excellent flux conservation property is retained and the new edge-based gradients evaluation significantly improves the robustness for high-aspect ratio meshes frequently encountered in three-dimensional, Navier-Stokes calculations. The order of accuracy of the proposed method is demonstrated for oblique acoustic wave propagation, shock-wave interaction, and hypersonic flows over a blunt body. The confirmed second-order convergence along with the enhanced robustness in handling hypersonic blunt body flow calculations makes the proposed approach a very competitive CFD framework for 3D Navier-Stokes simulations

The alpha(3) Scheme - A Fourth-Order Neutrally Stable CESE Solver

The conservation element and solution element (CESE) development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new 4th-order neutrally stable CESE solver of the advection equation Theta u/Theta + alpha Theta u/Theta x = 0. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and three points at the lower time level. Because it is associated with three independent mesh variables u(sup n) (sub j), (u(sub x))(sup n) (sub j) , and (uxz)(sup n) (sub j) (the numerical analogues of u, Theta u/Theta x, and Theta(exp 2)u/Theta x(exp 2), respectively) and four equations per mesh point, the new scheme is referred to as the alpha(3) scheme. As in the case of other similar CESE neutrally stable solvers, the alpha(3) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. These forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove that the alpha(3) scheme must be neutrally stable when it is stable. Moreover it is proved rigorously that all three amplification factors of the alpha(3) scheme are of unit magnitude for all phase angles if |v| <= 1/2 (v = alpha delta t/delta x). This theoretical result is consistent with the numerical stability condition |v| <= 1/2. Through numerical experiments, it is established that the alpha(3) scheme generally is (i) 4th-order accurate for the mesh variables u(sup n) (sub j) and (ux)(sup n) (sub j); and 2nd-order accurate for (uxx)(sup n) (sub j). However, in some exceptional cases, the scheme can achieve perfect accuracy aside from round-off errors

Simulation of Wave in Hypo-Elastic-Plastic Solids Modeled by Eulerian Conservation Laws

This paper reports a theoretical and numerical framework to model nonlinear waves in elastic-plastic solids. Formulated in the Eulerian frame, the governing equations employed include the continuity equation, the momentum equation, and an elastic-plastic constitutive relation. The complete governing equations are a set of first-order, fully coupled partial differential equations with source terms. The primary unknowns are velocities and deviatoric stresses. By casting the governing equations into a vector-matrix form, we derive the eigenvalues of the Jacobian matrix to show the wave speeds. The eigenvalues are also used to calculate the Courant number for numerical stability. The model equations are solved using the Space-Time Conservation Element and Solution Element (CESE) method. The approach is validated by comparing our numerical results to an analytical solution for the special case of longitudinal wave motion.Comment: 34 pages, 11 figure

Invariant conservative finite-difference schemes for the one-dimensional shallow water magnetohydrodynamics equations in Lagrangian coordinates

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.Comment: 18 pages, 6 figure

Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation

On the basis of the recent group classification of the one-dimensional magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction of symmetry-preserving finite-difference schemes with conservation laws is carried out. New schemes are constructed starting from the classical completely conservative Samarsky-Popov schemes. In the case of finite conductivity, schemes are derived that admit all the symmetries and possess all the conservation laws of the original differential model, including previously unknown conservation laws. In the case of a frozen-in magnetic field (when the conductivity is infinite), various schemes are constructed that possess conservation laws, including those preserving entropy along trajectories of motion. The peculiarities of constructing schemes with an extended set of conservation laws for specific forms of entropy and magnetic fluxes are discussed.Comment: 29 pages; some minor fixes and generalizations + Appendix containing an additional numerical schem