The computational complexity of traditional Lattice-Boltzmann methods for incompressible fluids

It is well-known that in fluid dynamics an alternative to customary direct solution methods (based on the discretization of the fluid fields) is provided by so-called \emph{particle simulation methods}. Particle simulation methods rely typically on appropriate \emph{kinetic models} for the fluid equations which permit the evaluation of the fluid fields in terms of suitable expectation values (or \emph{momenta}) of the kinetic distribution function $f(\mathbf{r,v},t),$ being respectively $\mathbf{r}$ and\textbf{\}$\mathbf{v}$ the position an velocity of a test particle with probability density $f(\mathbf{r,v},t)$. These kinetic models can be continuous or discrete in phase space, yielding respectively \emph{continuous} or \emph{discrete kinetic models} for the fluids. However, also particle simulation methods may be biased by an undesirable computational complexity. In particular, a fundamental issue is to estimate the algorithmic complexity of numerical simulations based on traditional LBM's (Lattice-Boltzmann methods; for review see Succi, 2001 \cite{Succi}). These methods, based on a discrete kinetic approach, represent currently an interesting alternative to direct solution methods. Here we intend to prove that for incompressible fluids fluids LBM's may present a high complexity. The goal of the investigation is to present a detailed account of the origin of the various complexity sources appearing in customary LBM's. The result is relevant to establish possible strategies for improving the numerical efficiency of existing numerical methods.Comment: Contributed paper at RGD26 (Kyoto, Japan, July 2008

A Parallel Mesh-Adaptive Framework for Hyperbolic Conservation Laws

We report on the development of a computational framework for the parallel, mesh-adaptive solution of systems of hyperbolic conservation laws like the time-dependent Euler equations in compressible gas dynamics or Magneto-Hydrodynamics (MHD) and similar models in plasma physics. Local mesh refinement is realized by the recursive bisection of grid blocks along each spatial dimension, implemented numerical schemes include standard finite-differences as well as shock-capturing central schemes, both in connection with Runge-Kutta type integrators. Parallel execution is achieved through a configurable hybrid of POSIX-multi-threading and MPI-distribution with dynamic load balancing. One- two- and three-dimensional test computations for the Euler equations have been carried out and show good parallel scaling behavior. The Racoon framework is currently used to study the formation of singularities in plasmas and fluids.Comment: late submissio

Numerical study of laminar magneto-convection in a differentially heated square duct

Magnetohydrodynamic pressure drops are one of the main issues for liquid metal blanket in fusion reactors. Minimize the fluid velocity at few millimeters per second is one strategy that can be employed to address the problem. For such low velocities, buoyant forces can effectively contribute to drive the flow and therefore must be considered in the blanket design. In order to do so, a CFD code able to represent magneto-convective phenomena is required. This work aims to gauge the capability of ANSYS© CFX-15 to solve such cases. The laminar flow in a differentially heated duct was selected as validation benchmark. A horizontal and uniform magnetic field was imposed over a square duct with a linear and constant temperature gradient perpendicular to the field. The fully developed flow was analyzed for Gr = 10^5 and Hartmann number (M) ranging from 10^2 to 10^3. Both insulating and conducting duct walls were considered. Strong dampening of the flow in the center of the duct was observed, whereas high velocity jets appeared close to the walls parallel to the magnetic field. The numerical results were validated against theoretical and numerical results founding an excellent agreement

A New Public Release of the GIZMO Code

We describe a major update to the public GIZMO code. GIZMO has been used in simulations of cosmology; galaxy and star formation and evolution; black hole accretion and feedback; proto-stellar disk dynamics and planet formation; fluid dynamics and plasma physics; dust-gas dynamics; giant impacts and solid-body interactions; collisionless gravitational dynamics; and more. This release of the public code supports: hydrodynamics (using various mesh-free finite-volume Godunov methods or SPH), ideal and non-ideal MHD, anisotropic conduction and viscosity, radiative cooling and chemistry, star and black hole formation and feedback, sink particles, dust-gas (aero)-dynamics (with or without magnetic fields), elastic/plastic dynamics, arbitrary (gas, stellar, degenerate, solid/liquid material) equations of state, passive scalar/turbulent diffusion, large-eddy and shearing boxes, self-gravity with fully-adaptive force softenings, arbitrary cosmological expansion, and on-the-fly group-finding. It is massively-parallel with hybrid MPI+OpenMP scaling verified up to >1 million threads. The code is extensively documented, with test problems and tutorials provided for these different physics modules.Comment: Brief (2 page) overview. The GIZMO code (with an extensive User Guide, animations, and test problems) is available through http://www.tapir.caltech.edu/~phopkins/Site/GIZMO.html or on the repository at https://bitbucket.org/phopkins/gizmo-publi

"Magneto-elastic" waves in an anisotropic magnetised plasma

The linear waves that propagate in a two fluid magnetised plasma allowing for a non-gyrotropic perturbed ion pressure tensor are investigated. For perpendicular propagation and perturbed fluid velocity a low frequency (magnetosonic) and a high frequency (ion Bernstein) branch are identified and discussed. For both branches a comparison is made with the results of a truncated Vlasov treatment. For the low frequency branch we show that a consistent expansion procedure allows us to recover the correct expression of the Finite Larmor Radius corrections to the magnetosonic dispersion relation.Comment: 16 pages, 9 figure

FISH: A 3D parallel MHD code for astrophysical applications

FISH is a fast and simple ideal magneto-hydrodynamics code that scales to ~10 000 processes for a Cartesian computational domain of ~1000^3 cells. The simplicity of FISH has been achieved by the rigorous application of the operator splitting technique, while second order accuracy is maintained by the symmetric ordering of the operators. Between directional sweeps, the three-dimensional data is rotated in memory so that the sweep is always performed in a cache-efficient way along the direction of contiguous memory. Hence, the code only requires a one-dimensional description of the conservation equations to be solved. This approach also enable an elegant novel parallelisation of the code that is based on persistent communications with MPI for cubic domain decomposition on machines with distributed memory. This scheme is then combined with an additional OpenMP parallelisation of different sweeps that can take advantage of clusters of shared memory. We document the detailed implementation of a second order TVD advection scheme based on flux reconstruction. The magnetic fields are evolved by a constrained transport scheme. We show that the subtraction of a simple estimate of the hydrostatic gradient from the total gradients can significantly reduce the dissipation of the advection scheme in simulations of gravitationally bound hydrostatic objects. Through its simplicity and efficiency, FISH is as well-suited for hydrodynamics classes as for large-scale astrophysical simulations on high-performance computer clusters. In preparation for the release of a public version, we demonstrate the performance of FISH in a suite of astrophysically orientated test cases.Comment: 27 pages, 11 figure

A Simflowny-based high-performance 3D code for the generalized induction equation

In the interior of neutron stars, the induction equation regulates the long-term evolution of the magnetic fields by means of resistivity, Hall dynamics and ambipolar diffusion. Despite the apparent simplicity and compactness of the equation, the dynamics it describes is not trivial and its understanding relies on accurate numerical simulations. While a few works in 2D have reached a mature stage and a consensus on the general dynamics at least for some simple initial data, only few attempts have been performed in 3D, due to the computational costs and the need for a proper numerical treatment of the intrinsic non-linearity of the equation. Here, we carefully analyze the general induction equation, studying its characteristic structure, and we present a new Cartesian 3D code, generated by the user-friendly, publicly available {\em Simflowny} platform. The code uses high-order numerical schemes for the time and spatial discretization, and relies on the highly-scalable {\em SAMRAI} architecture for the adaptive mesh refinement. We present the application of the code to several benchmark tests, showing the high order of convergence and accuracy achieved and the capabilities in terms of magnetic shock resolution and three-dimensionality. This paper paves the way for the applications to a realistic, 3D long-term evolution of neutron stars interior and, possibly, of other astrophysical sources.Comment: 23 pages, 13 figures. In pres

Development of a computational model for predicting solar wind flows past nonmagnetic terrestrial planets

A computational model for the determination of the detailed plasma and magnetic field properties of the global interaction of the solar wind with nonmagnetic terrestrial planetary obstacles is described. The theoretical method is based on an established single fluid, steady, dissipationless, magnetohydrodynamic continuum model, and is appropriate for the calculation of supersonic, super-Alfvenic solar wind flow past terrestrial ionospheres