208 research outputs found
A Two-dimensional HLLC Riemann Solver for Conservation Laws : Application to Euler and MHD Flows
In this paper we present a genuinely two-dimensional HLLC Riemann solver. On
logically rectangular meshes, it accepts four input states that come together
at an edge and outputs the multi-dimensionally upwinded fluxes in both
directions. This work builds on, and improves, our prior work on
two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here
achieves its stabilization by introducing a constant state in the region of
strong interaction, where four one-dimensional Riemann problems interact
vigorously with one another. A robust version of the HLL Riemann solver is
presented here along with a strategy for introducing sub-structure in the
strongly-interacting state. Introducing sub-structure turns the two-dimensional
HLL Riemann solver into a two-dimensional HLLC Riemann solver. The
sub-structure that we introduce represents a contact discontinuity which can be
oriented in any direction relative to the mesh.
The Riemann solver presented here is general and can work with any system of
conservation laws. We also present a second order accurate Godunov scheme that
works in three dimensions and is entirely based on the present multidimensional
HLLC Riemann solver technology. The methods presented are cost-competitive with
traditional higher order Godunov schemes
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
A multidimensional grid-adaptive relativistic magnetofluid code
A robust second order, shock-capturing numerical scheme for multi-dimensional
special relativistic magnetohydrodynamics on computational domains with
adaptive mesh refinement is presented. The base solver is a total variation
diminishing Lax-Friedrichs scheme in a finite volume setting and is combined
with a diffusive approach for controlling magnetic monopole errors. The
consistency between the primitive and conservative variables is ensured at all
limited reconstructions and the spatial part of the four velocity is used as a
primitive variable. Demonstrative relativistic examples are shown to validate
the implementation. We recover known exact solutions to relativistic MHD
Riemann problems, and simulate the shock-dominated long term evolution of
Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication
A Second Order Godunov Method for Multidimensional Relativistic Magnetohydrodynamics
We describe a new Godunov algorithm for relativistic magnetohydrodynamics
(RMHD) that combines a simple, unsplit second order accurate integrator with
the constrained transport (CT) method for enforcing the solenoidal constraint
on the magnetic field. A variety of approximate Riemann solvers are implemented
to compute the fluxes of the conserved variables. The methods are tested with a
comprehensive suite of multidimensional problems. These tests have helped us
develop a hierarchy of correction steps that are applied when the integration
algorithm predicts unphysical states due to errors in the fluxes, or errors in
the inversion between conserved and primitive variables. Although used
exceedingly rarely, these corrections dramatically improve the stability of the
algorithm. We present preliminary results from the application of these
algorithms to two problems in RMHD: the propagation of supersonic magnetized
jets, and the amplification of magnetic field by turbulence driven by the
relativistic Kelvin-Helmholtz instability (KHI). Both of these applications
reveal important differences between the results computed with Riemann solvers
that adopt different approximations for the fluxes. For example, we show that
use of Riemann solvers which include both contact and rotational
discontinuities can increase the strength of the magnetic field within the
cocoon by a factor of ten in simulations of RMHD jets, and can increase the
spectral resolution of three-dimensional RMHD turbulence driven by the KHI by a
factor of 2. This increase in accuracy far outweighs the associated increase in
computational cost. Our RMHD scheme is publicly available as part of the Athena
code.Comment: 75 pages, 28 figures, accepted for publication in ApJS. Version with
high resolution figures available from
http://jila.colorado.edu/~krb3u/Athena_SR/rmhd_method_paper.pd
Relativistic MHD Simulations of Jets with Toroidal Magnetic Fields
This paper presents an application of the recent relativistic HLLC
approximate Riemann solver by Mignone & Bodo to magnetized flows with vanishing
normal component of the magnetic field.
The numerical scheme is validated in two dimensions by investigating the
propagation of axisymmetric jets with toroidal magnetic fields.
The selected jet models show that the HLLC solver yields sharper resolution
of contact and shear waves and better convergence properties over the
traditional HLL approach.Comment: 12 pages, 5 figure
Relativistic Magnetohydrodynamics: Renormalized eigenvectors and full wave decomposition Riemann solver
We obtain renormalized sets of right and left eigenvectors of the flux vector
Jacobians of the relativistic MHD equations, which are regular and span a
complete basis in any physical state including degenerate ones. The
renormalization procedure relies on the characterization of the degeneracy
types in terms of the normal and tangential components of the magnetic field to
the wavefront in the fluid rest frame. Proper expressions of the renormalized
eigenvectors in conserved variables are obtained through the corresponding
matrix transformations. Our work completes previous analysis that present
different sets of right eigenvectors for non-degenerate and degenerate states,
and can be seen as a relativistic generalization of earlier work performed in
classical MHD. Based on the full wave decomposition (FWD) provided by the the
renormalized set of eigenvectors in conserved variables, we have also developed
a linearized (Roe-type) Riemann solver. Extensive testing against one- and
two-dimensional standard numerical problems allows us to conclude that our
solver is very robust. When compared with a family of simpler solvers that
avoid the knowledge of the full characteristic structure of the equations in
the computation of the numerical fluxes, our solver turns out to be less
diffusive than HLL and HLLC, and comparable in accuracy to the HLLD solver. The
amount of operations needed by the FWD solver makes it less efficient
computationally than those of the HLL family in one-dimensional problems.
However its relative efficiency increases in multidimensional simulations.Comment: 50 pages, 17 figures (2 in color). Submitted to ApJ Suppl. Se
An HLLC Solver for Relativistic Flows -- II. Magnetohydrodynamics
An approximate Riemann solver for the equations of relativistic
magnetohydrodynamics (RMHD) is derived. The HLLC solver, originally developed
by Toro, Spruce and Spears, generalizes the algorithm described in a previous
paper (Mignone & Bodo 2004) to the case where magnetic fields are present. The
solution to the Riemann problem is approximated by two constant states bounded
by two fast shocks and separated by a tangential wave. The scheme is
Jacobian-free, in the sense that it avoids the expensive characteristic
decomposition of the RMHD equations and it improves over the HLL scheme by
restoring the missing contact wave.
Multidimensional integration proceeds via the single step, corner transport
upwind (CTU) method of Colella, combined with the contrained tranport (CT)
algorithm to preserve divergence-free magnetic fields. The resulting numerical
scheme is simple to implement, efficient and suitable for a general equation of
state. The robustness of the new algorithm is validated against one and two
dimensional numerical test problems.Comment: 17 pages, 12 figure
Astrophysical Weighted Particle Magnetohydrodynamics
This paper presents applications of weighted meshless scheme for conservation
laws to the Euler equations and the equations of ideal magnetohydrodynamics.
The divergence constraint of the latter is maintained to the truncation error
by a new meshless divergence cleaning procedure. The physics of the interaction
between the particles is described by an one-dimensional Riemann problem in a
moving frame. As a result, necessary diffusion which is required to treat
dissipative processes is added automatically. As a result, our scheme has no
free parameters that controls the physics of inter-particle interaction, with
the exception of the number of the interacting neighbours which control the
resolution and accuracy. The resulting equations have the form similar to SPH
equations, and therefore existing SPH codes can be used to implement the
weighed particle scheme. The scheme is validated in several hydrodynamic and
MHD test cases. In particular, we demonstrate for the first time the ability of
a meshless MHD scheme to model magneto-rotational instability in accretion
disks.Comment: 27 pages, 24 figures, 1 column, submitted to MNRAS, hi-res version
can be obtained at http://www.strw.leidenuniv.nl/~egaburov/wpmhd.pd
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