3,536 research outputs found
A fast immersed boundary method for external incompressible viscous flows using lattice Green's functions
A new parallel, computationally efficient immersed boundary method for
solving three-dimensional, viscous, incompressible flows on unbounded domains
is presented. Immersed surfaces with prescribed motions are generated using the
interpolation and regularization operators obtained from the discrete delta
function approach of the original (Peskin's) immersed boundary method. Unlike
Peskin's method, boundary forces are regarded as Lagrange multipliers that are
used to satisfy the no-slip condition. The incompressible Navier-Stokes
equations are discretized on an unbounded staggered Cartesian grid and are
solved in a finite number of operations using lattice Green's function
techniques. These techniques are used to automatically enforce the natural
free-space boundary conditions and to implement a novel block-wise adaptive
grid that significantly reduces the run-time cost of solutions by limiting
operations to grid cells in the immediate vicinity and near-wake region of the
immersed surface. These techniques also enable the construction of practical
discrete viscous integrating factors that are used in combination with
specialized half-explicit Runge-Kutta schemes to accurately and efficiently
solve the differential algebraic equations describing the discrete momentum
equation, incompressibility constraint, and no-slip constraint. Linear systems
of equations resulting from the time integration scheme are efficiently solved
using an approximation-free nested projection technique. The algebraic
properties of the discrete operators are used to reduce projection steps to
simple discrete elliptic problems, e.g. discrete Poisson problems, that are
compatible with recent parallel fast multipole methods for difference
equations. Numerical experiments on low-aspect-ratio flat plates and spheres at
Reynolds numbers up to 3,700 are used to verify the accuracy and physical
fidelity of the formulation.Comment: 32 pages, 9 figures; preprint submitted to Journal of Computational
Physic
Relativistic MHD and black hole excision: Formulation and initial tests
A new algorithm for solving the general relativistic MHD equations is
described in this paper. We design our scheme to incorporate black hole
excision with smooth boundaries, and to simplify solving the combined Einstein
and MHD equations with AMR. The fluid equations are solved using a finite
difference Convex ENO method. Excision is implemented using overlapping grids.
Elliptic and hyperbolic divergence cleaning techniques allow for maximum
flexibility in choosing coordinate systems, and we compare both methods for a
standard problem. Numerical results of standard test problems are presented in
two-dimensional flat space using excision, overlapping grids, and elliptic and
hyperbolic divergence cleaning.Comment: 22 pages, 8 figure
Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity
Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft
A Second-Order Unsplit Godunov Scheme for Cell-Centered MHD: the CTU-GLM scheme
We assess the validity of a single step Godunov scheme for the solution of
the magneto-hydrodynamics equations in more than one dimension. The scheme is
second-order accurate and the temporal discretization is based on the
dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The
proposed scheme employs a cell-centered representation of the primary fluid
variables (including magnetic field) and conserves mass, momentum, magnetic
induction and energy. A variant of the scheme, which breaks momentum and energy
conservation, is also considered. Divergence errors are transported out of the
domain and damped using the mixed hyperbolic/parabolic divergence cleaning
technique by Dedner et al. (J. Comput. Phys., 175, 2002). The strength and
accuracy of the scheme are verified by a direct comparison with the eight-wave
formulation (also employing a cell-centered representation) and with the
popular constrained transport method, where magnetic field components retain a
staggered collocation inside the computational cell. Results obtained from two-
and three-dimensional test problems indicate that the newly proposed scheme is
robust, accurate and competitive with recent implementations of the constrained
transport method while being considerably easier to implement in existing hydro
codes.Comment: 31 Pages, 16 Figures Accepted for publication in Journal of
Computational Physic
GRMHD in axisymmetric dynamical spacetimes: the X-ECHO code
We present a new numerical code, X-ECHO, for general relativistic
magnetohydrodynamics (GRMHD) in dynamical spacetimes. This is aimed at studying
astrophysical situations where strong gravity and magnetic fields are both
supposed to play an important role, such as for the evolution of magnetized
neutron stars or for the gravitational collapse of the magnetized rotating
cores of massive stars, which is the astrophysical scenario believed to
eventually lead to (long) GRB events. The code is based on the extension of the
Eulerian conservative high-order (ECHO) scheme [Del Zanna et al., A&A 473, 11
(2007)] for GRMHD, here coupled to a novel solver for the Einstein equations in
the extended conformally flat condition (XCFC). We fully exploit the 3+1
Eulerian formalism, so that all the equations are written in terms of familiar
3D vectors and tensors alone, we adopt spherical coordinates for the conformal
background metric, and we consider axisymmetric spacetimes and fluid
configurations. The GRMHD conservation laws are solved by means of
shock-capturing methods within a finite-difference discretization, whereas, on
the same numerical grid, the Einstein elliptic equations are treated by
resorting to spherical harmonics decomposition and solved, for each harmonic,
by inverting band diagonal matrices. As a side product, we build and make
available to the community a code to produce GRMHD axisymmetric equilibria for
polytropic relativistic stars in the presence of differential rotation and a
purely toroidal magnetic field. This uses the same XCFC metric solver of the
main code and has been named XNS. Both XNS and the full X-ECHO codes are
validated through several tests of astrophysical interest.Comment: 18 pages, 9 figures, accepted for publication in A&
An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics
In plasma simulations, where the speed of light divided by a characteristic
length is at a much higher frequency than other relevant parameters in the
underlying system, such as the plasma frequency, implicit methods begin to play
an important role in generating efficient solutions in these multi-scale
problems. Under conditions of scale separation, one can rescale Maxwell's
equations in such a way as to give a magneto static limit known as the Darwin
approximation of electromagnetics. In this work, we present a new approach to
solve Maxwell's equations based on a Method of Lines Transpose (MOL)
formulation, combined with a fast summation method with computational
complexity , where is the number of grid points (particles).
Under appropriate scaling, we show that the proposed schemes result in
asymptotic preserving methods that can recover the Darwin limit of
electrodynamics
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