113 research outputs found
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
High-order conservative reconstruction schemes for finite volume methods in cylindrical and spherical coordinates
High-order reconstruction schemes for the solution of hyperbolic conservation
laws in orthogonal curvilinear coordinates are revised in the finite volume
approach. The formulation employs a piecewise polynomial approximation to the
zone-average values to reconstruct left and right interface states from within
a computational zone to arbitrary order of accuracy by inverting a
Vandermonde-like linear system of equations with spatially varying
coefficients. The approach is general and can be used on uniform and
non-uniform meshes although explicit expressions are derived for polynomials
from second to fifth degree in cylindrical and spherical geometries with
uniform grid spacing. It is shown that, in regions of large curvature, the
resulting expressions differ considerably from their Cartesian counterparts and
that the lack of such corrections can severely degrade the accuracy of the
solution close to the coordinate origin. Limiting techniques and monotonicity
constraints are revised for conventional reconstruction schemes, namely, the
piecewise linear method (PLM), third-order weighted essentially non-oscillatory
(WENO) scheme and the piecewise parabolic method (PPM).
The performance of the improved reconstruction schemes is investigated in a
number of selected numerical benchmarks involving the solution of both scalar
and systems of nonlinear equations (such as the equations of gas dynamics and
magnetohydrodynamics) in cylindrical and spherical geometries in one and two
dimensions. Results confirm that the proposed approach yields considerably
smaller errors, higher convergence rates and it avoid spurious numerical
effects at a symmetry axis.Comment: 37 pages, 12 Figures. Accepted for publication in Journal of
Compuational Physic
The Athena Astrophysical MHD Code in Cylindrical Geometry
A method for implementing cylindrical coordinates in the Athena
magnetohydrodynamics (MHD) code is described. The extension follows the
approach of Athena's original developers and has been designed to alter the
existing Cartesian-coordinates code as minimally and transparently as possible.
The numerical equations in cylindrical coordinates are formulated to maintain
consistency with constrained transport, a central feature of the Athena
algorithm, while making use of previously implemented code modules such as the
Riemann solvers. Angular-momentum transport, which is critical in astrophysical
disk systems dominated by rotation, is treated carefully. We describe
modifications for cylindrical coordinates of the higher-order spatial
reconstruction and characteristic evolution steps as well as the finite-volume
and constrained transport updates. Finally, we present a test suite of standard
and novel problems in one-, two-, and three-dimensions designed to validate our
algorithms and implementation and to be of use to other code developers. The
code is suitable for use in a wide variety of astrophysical applications and is
freely available for download on the web
Piecewise Parabolic Method on a Local Stencil for Magnetized Supersonic Turbulence Simulation
Stable, accurate, divergence-free simulation of magnetized supersonic
turbulence is a severe test of numerical MHD schemes and has been surprisingly
difficult to achieve due to the range of flow conditions present. Here we
present a new, higher order-accurate, low dissipation numerical method which
requires no additional dissipation or local "fixes" for stable execution. We
describe PPML, a local stencil variant of the popular PPM algorithm for solving
the equations of compressible ideal magnetohydrodynamics. The principal
difference between PPML and PPM is that cell interface states are evolved
rather that reconstructed at every timestep, resulting in a compact stencil.
Interface states are evolved using Riemann invariants containing all transverse
derivative information. The conservation laws are updated in an unsplit
fashion, making the scheme fully multidimensional. Divergence-free evolution of
the magnetic field is maintained using the higher order-accurate constrained
transport technique of Gardiner and Stone. The accuracy and stability of the
scheme is documented against a bank of standard test problems drawn from the
literature. The method is applied to numerical simulation of supersonic MHD
turbulence, which is important for many problems in astrophysics, including
star formation in dark molecular clouds. PPML accurately reproduces in
three-dimensions a transition to turbulence in highly compressible isothermal
gas in a molecular cloud model. The low dissipation and wide spectral bandwidth
of this method make it an ideal candidate for direct turbulence simulations.Comment: 28 pages, 18 figure
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