70 research outputs found
von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers
The time-dependent equations of computational electrodynamics (CED) are
evolved consistent with the divergence constraints. As a result, there has been
a recent effort to design finite volume time domain (FVTD) and discontinuous
Galerkin time domain (DGTD) schemes that satisfy the same constraints and,
nevertheless, draw on recent advances in higher order Godunov methods. This
paper catalogues the first step in the design of globally constraint-preserving
DGTD schemes. The algorithms presented here are based on a novel DG-like method
that is applied to a Yee-type staggering of the electromagnetic field variables
in the faces of the mesh. The other two novel building blocks of the method
include constraint-preserving reconstruction of the electromagnetic fields and
multidimensional Riemann solvers; both of which have been developed in recent
years by the first author. We carry out a von Neumann stability analysis of the
entire suite of DGTD schemes for CED at orders of accuracy ranging from second
to fourth. A von Neumann stability analysis gives us the maximal CFL numbers
that can be sustained by the DGTD schemes presented here at all orders. It also
enables us to understand the wave propagation characteristics of the schemes in
various directions on a Cartesian mesh. We find that the CFL of DGTD schemes
decreases with increasing order. To counteract that, we also present
constraint-preserving PNPM schemes for CED. We find that the third and fourth
order constraint-preserving DGTD and P1PM schemes have some extremely
attractive properties when it comes to low-dispersion, low-dissipation
propagation of electromagnetic waves in multidimensions. Numerical accuracy
tests are also provided to support the von Neumann stability analysis
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
The Evolution of Adiabatic Supernova Remnants in a Turbulent, Magnetized Medium
(Abridged) We present the results of three dimensional calculations for the
MHD evolution of an adiabatic supernova remnant in both a uniform and turbulent
interstellar medium using the RIEMANN framework of Balsara. In the uniform
case, which contains an initially uniform magnetic field, the density structure
of the shell remains largely spherical, while the magnetic pressure and
synchrotron emissivity are enhanced along the plane perpendicular to the field
direction. This produces a bilateral or barrel-type morphology in synchrotron
emission for certain viewing angles. We then consider a case with a turbulent
external medium as in Balsara & Pouquet, characterized by .
Several important changes are found. First, despite the presence of a uniform
field, the overall synchrotron emissivity becomes approximately spherically
symmetric, on the whole, but is extremely patchy and time-variable, with
flickering on the order of a few computational time steps. We suggest that the
time and spatial variability of emission in early phase SNR evolution provides
information on the turbulent medium surrounding the remnant. The
shock-turbulence interaction is also shown to be a strong source of
helicity-generation and, therefore, has important consequences for magnetic
field generation. We compare our calculations to the Sedov-phase evolution, and
discuss how the emission characteristics of SNR may provide a diagnostic on the
nature of turbulence in the pre-supernova environment.Comment: ApJ, in press, 5 color figure
An Intercomparison Between Divergence-Cleaning and Staggered Mesh Formulations for Numerical Magnetohydrodynamics
In recent years, several different strategies have emerged for evolving the
magnetic field in numerical MHD. Some of these methods can be classified as
divergence-cleaning schemes, where one evolves the magnetic field components
just like any other variable in a higher order Godunov scheme. The fact that
the magnetic field is divergence-free is imposed post-facto via a
divergence-cleaning step. Other schemes for evolving the magnetic field rely on
a staggered mesh formulation which is inherently divergence-free. The claim has
been made that the two approaches are equivalent. In this paper we
cross-compare three divergence-cleaning schemes based on scalar and vector
divergence-cleaning and a popular divergence-free scheme. All schemes are
applied to the same stringent test problem. Several deficiencies in all the
divergence-cleaning schemes become clearly apparent with the scalar
divergence-cleaning schemes performing worse than the vector
divergence-cleaning scheme. The vector divergence-cleaning scheme also shows
some deficiencies relative to the staggered mesh divergence-free scheme. The
differences can be explained by realizing that all the divergence-cleaning
schemes are based on a Poisson solver which introduces a non-locality into the
scheme, though other subtler points of difference are also catalogued. By using
several diagnostics that are routinely used in the study of turbulence, it is
shown that the differences in the schemes produce measurable differences in
physical quantities that are of interest in such studies
ADER-WENO Finite Volume Schemes with Space-Time Adaptive Mesh Refinement
We present the first high order one-step ADER-WENO finite volume scheme with
Adaptive Mesh Refinement (AMR) in multiple space dimensions. High order spatial
accuracy is obtained through a WENO reconstruction, while a high order one-step
time discretization is achieved using a local space-time discontinuous Galerkin
predictor method. Due to the one-step nature of the underlying scheme, the
resulting algorithm is particularly well suited for an AMR strategy on
space-time adaptive meshes, i.e.with time-accurate local time stepping. The AMR
property has been implemented 'cell-by-cell', with a standard tree-type
algorithm, while the scheme has been parallelized via the Message Passing
Interface (MPI) paradigm. The new scheme has been tested over a wide range of
examples for nonlinear systems of hyperbolic conservation laws, including the
classical Euler equations of compressible gas dynamics and the equations of
magnetohydrodynamics (MHD). High order in space and time have been confirmed
via a numerical convergence study and a detailed analysis of the computational
speed-up with respect to highly refined uniform meshes is also presented. We
also show test problems where the presented high order AMR scheme behaves
clearly better than traditional second order AMR methods. The proposed scheme
that combines for the first time high order ADER methods with space--time
adaptive grids in two and three space dimensions is likely to become a useful
tool in several fields of computational physics, applied mathematics and
mechanics.Comment: With updated bibliography informatio
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