A new code for astrophysical magnetohydrodynamics (MHD) is described. The code has been designed to be easily extensible for use with static and adaptive mesh refinement. It combines higher order Godunov methods with the constrained transport (CT) technique to enforce the divergence-free constraint on the magnetic field. Discretization is based on cell-centered volume averages for mass, momentum, and energy, and face-centered area averages for the magnetic field. Novel features of the algorithm include (1) a consistent framework for computing the time- and edge-averaged electric fields used by CT to evolve the magnetic field from the time- and area-averaged Godunov fluxes, (2) the extension to MHD of spatial reconstruction schemes that involve a dimensionally split time advance, and (3) the extension to MHD of two different dimensionally unsplit integration methods. Implementation of the algorithm in both C and FORTRAN95 is detailed, including strategies for parallelization using domain decomposition. Results from a test suite which includes problems in one-, two-, and three-dimensions for both hydrodynamics and MHD are given, not only to demonstrate the fidelity of the algorithms, but also to enable comparisons to other methods. The source code is freely available for download on the web.

/pdf/athena-a-new-code-for-astrophysical-mhd-1de7wtkkl8.pdf

Athena: a new code for astrophysical mhd

https://ntrs.nasa.gov/api/citations/20110005631/downloads/20110005631.pdf

Adaptive numerical algorithms in space weather modeling

A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics

https://users.monash.edu.au/~dprice/ndspmhd/price-spmhd.pdf

Smoothed particle hydrodynamics and magnetohydrodynamics

A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

http://www.tp4.rub.de/~sw/papers/JCP2002.6961.pdf

Hyperbolic divergence cleaning for the MHD equations

The methods most frequently used in computational fluid mechanics for solving the compressible Navier-Stokes or compressible Euler equations are finite volume schemes on structured or on unstructured grids. First order as well as higher order space discretizations of MUSCL type, including flux limiters and higher order Runge- Kutta methods for the time discretization, guarantee robust and accurate schemes. But there is an important difficulty. If one increases the order, the stencil for the space discretization increases too, and the scheme becomes very expensive. Therefore schemes with more compact stencils are necessary. Discontinuous Galerkin schemes in the sense of [3] are of this type. They are identical to finite volume schemes in the case of formal first order, and for higher order they use nonconformal ansatz functions whose restrictions to single cells are polynomials of higher order. Therefore they seem to be more efficient and it is of highest interest to compare finite volume and discontinuous Galerkin methods for real applications with respect to their efficiency. Experiences [1] with the Euler equations of gas dynamics indicate that the discontinuous Galerkin methods have some advantages. Since there are no systematic studies available in the literature, we will present in this paper some numerical experiments for hyperbolic conservation laws in multiple space dimensions to compare their efficiency for different situations. As important instances of hyperbolic conservation laws we consider the Euler equations of gas dynamics and Lundquist’s equations of ideal magneto-hydrodynamics (MHD). Furthermore we have found a new limiter which improves the results from [14]. Similar studies have been done in [4].

Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions

PAIN publishes research on the nature, mechanisms and treatment of pain. The journal provides a forum for the dissemination of research in the basic and clinical sciences of multidisciplinary interest. 

T. Schnitzer

Papers

Hyperbolic divergence cleaning for the MHD equations

Comparison of Finite Volume and Discontinuous Galerkin Methods of Higher Order for Systems of Conservation Laws in Multiple Space Dimensions

Chronic pain domains and their relationship to personality, abilities, and brain networks.