15,488 research outputs found
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
Numerical Methods for the Stochastic Landau-Lifshitz Navier-Stokes Equations
The Landau-Lifshitz Navier-Stokes (LLNS) equations incorporate thermal
fluctuations into macroscopic hydrodynamics by using stochastic fluxes. This
paper examines explicit Eulerian discretizations of the full LLNS equations.
Several CFD approaches are considered (including MacCormack's two-step
Lax-Wendroff scheme and the Piecewise Parabolic Method) and are found to give
good results (about 10% error) for the variances of momentum and energy
fluctuations. However, neither of these schemes accurately reproduces the
density fluctuations. We introduce a conservative centered scheme with a
third-order Runge-Kutta temporal integrator that does accurately produce
density fluctuations. A variety of numerical tests, including the random walk
of a standing shock wave, are considered and results from the stochastic LLNS
PDE solver are compared with theory, when available, and with molecular
simulations using a Direct Simulation Monte Carlo (DSMC) algorithm
Local time steps for a finite volume scheme
We present a strategy for solving time-dependent problems on grids with local
refinements in time using different time steps in different regions of space.
We discuss and analyze two conservative approximations based on finite volume
with piecewise constant projections and domain decomposition techniques. Next
we present an iterative method for solving the composite-grid system that
reduces to solution of standard problems with standard time stepping on the
coarse and fine grids. At every step of the algorithm, conservativity is
ensured. Finally, numerical results illustrate the accuracy of the proposed
methods
Flux form Semi-Lagrangian methods for parabolic problems
A semi-Lagrangian method for parabolic problems is proposed, that extends
previous work by the authors to achieve a fully conservative, flux-form
discretization of linear and nonlinear diffusion equations. A basic consistency
and convergence analysis are proposed. Numerical examples validate the proposed
method and display its potential for consistent semi-Lagrangian discretization
of advection--diffusion and nonlinear parabolic problems
- …