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

Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows

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 pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow. We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical flows. A robust and efficient second order accurate numerical scheme for two and three dimensional Euler and magnetohydrodynamic flows is presented. The scheme is built on the current multidimensional Riemann solver. The number of zones updated per second by this scheme on a modern processor is shown to be cost competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps

Numerical simulation of unsteady MHD flows and applications

International audienceWe present a robust numerical method for solving the compressible Ideal Magneto-Hydrodynamic equations. It is based on the Residual Distribution (RD) algorithms already successfully tested in many problems. We adapted the scheme to the multi-dimensional unsteady MHD model. The constraint ∇ · B = 0 is enforced by the use a Generalized Lagrange Multiplier (GLM) technique. First, we present this complete system and the keys to get its eigensystem, as we may need it in the algorithm. Next, we introduce the numerical scheme built in order to get a compressible, unsteady and implicit solver which has good shock-capturing properties and is second-order accurate at the converged state. To show the efficiency of our method, we will then comment some 2D results. We will end by pointing out some issues and the extensions we plan for this solver

A parallel solution-adaptive scheme for ideal magnetohydrodynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77232/1/AIAA-1999-3273-200.pd

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.Comment: 32 pages, 14 figure, submitted to Journal of Computational Physics (Aug 7 2009

An Unsplit, Cell-Centered Godunov Method for Ideal MHD

We present a second-order Godunov algorithm for multidimensional, ideal MHD. Our algorithm is based on the unsplit formulation of Colella (J. Comput. Phys. vol. 87, 1990), with all of the primary dependent variables centered at the same location. To properly represent the divergence-free condition of the magnetic fields, we apply a discrete projection to the intermediate values of the field at cell faces, and apply a filter to the primary dependent variables at the end of each time step. We test the method against a suite of linear and nonlinear tests to ascertain accuracy and stability of the scheme under a variety of conditions. The test suite includes rotated planar linear waves, MHD shock tube problems, low-beta flux tubes, and a magnetized rotor problem. For all of these cases, we observe that the algorithm is second-order accurate for smooth solutions, converges to the correct weak solution for problems involving shocks, and exhibits no evidence of instability or loss of accuracy due to the possible presence of non-solenoidal fields.Comment: 37 Pages, 9 Figures, submitted to Journal of Computational Physic