High‐Order Upwind Schemes for Multidimensional Magnetohydrodynamics

Abstract

A general method for constructing high-order upwind schemes for multidimensional magnetohydrodynamics (MHD), having as a main built-in condition the divergence-free constraint $ AE B \ 0 for the magnetic Ðeld vector B, is proposed. The suggested procedure is based on consistency arguments, by taking into account the speciÐc operator structure of MHD equations with respect to the reference Euler equations of gasdynamics. This approach leads in a natural way to a staggered representation of the B Ðeld numerical data in which the divergence-free condition in the cell-averaged form, corresponding to second-order accurate numerical derivatives, is exactly fulÐlled. To extend this property to higher order schemes, we then give general prescriptions to satisfy a (r ] 1)th order accurate $ AE B \ 0 relation for any numerical B Ðeld having a rth order interpolation accuracy. Consistency arguments lead also to a proper formulation of the upwind procedures needed to integrate the induction equations, assuring the exact conservation in time of the divergence-free condition and the related continuity properties for the B vector components. As an application, a third-order code to simulate multidimensional MHD Ñows of astrophysical interest is developed using essentially nonoscillatoryÈbased reconstruction algorithms. Several test problems to illustrate and validate the proposed approach are Ðnally presented.

An efficient shock-capturing central-type scheme for multidimensional relativistic flows

“…The central shock-capturing scheme of Londrillo & Del Zanna (2000) is extended to the relativistic case, here for ideal RHD flows and to RMHD in the next paper of the series. Compared to other schemes proposed for relativistic astrophysical problems over the last decade in the literature, our method is extremely simple and efficient, since no eigenvector decomposition and Riemann solvers are involved.…”

“…However, due to the high resolution of ENO methods, characteristic decomposition on every point of the interpolation stencil becomes prohibitive when moving to three-dimensional simulations, especially for relativistic flows where Jacobian matrices are more complex than in the Eulerian case. The same kind of problem has to be faced in magnetohydrodynamics (MHD, see Londrillo & Del Zanna 2000, from now on LD), because of the increasing number of variables, equations and eigenmodes. The worst possible case is obviously that of relativistic MHD, for which only second Following the scheme proposed in LD, in the present series of two papers a shock-capturing scheme that avoids the expensive characteristic decomposition, still retaining nonoscillatory properties, will be suggested, first for ideal relativistic hydrodynamics (RHD, in this paper) and then for ideal relativistic magnetohydrodynamics (RMHD, in the next paper of the series).…”

Divergence-Free Adaptive Mesh Refinement for Magnetohydrodynamics

“…7) An electric field correction strategy is presented which restores the consistency of electric fields at a fine-coarse interface in the AMR hierarchy. 34 8) Because of the above four points, the time-step can be sub-cycled on finer meshes without loss of the divergence-free property of the magnetic fields. Points 4) and 5) are also essential for the time-step sub-cycling because they provide for a robust and stable numerical strategy.…”

High-order Godunov schemes for global 3D MHD simulations of accretion disks

“…For a staggered field, the one-dimensional solutions of the Riemann problem for density, momentum and the energy equation have no direct extension to the upwind fluxes in the induction equation (Balsara & Spicer 1999). Londrillo & Del Zanna (2000) and Londrillo & del Zanna (2004) enhanced the CT method to make it consistent with the one-dimensional solver for plane parallel, grid-aligned flows. Based on the Harten-Lax-van Leer (HLL) and Roe Riemann solver fluxes, they proposed a new way to reconstruct the electric fields now called "upwind CT".…”