166 research outputs found
Variational Integrators for Reduced Magnetohydrodynamics
Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics
equations with applications to both fusion and astrophysical plasmas,
possessing a noncanonical Hamiltonian structure and consequently a number of
conserved functionals. We propose a new discretisation strategy for these
equations based on a discrete variational principle applied to a formal
Lagrangian. The resulting integrator preserves important quantities like the
total energy, magnetic helicity and cross helicity exactly (up to machine
precision). As the integrator is free of numerical resistivity, spurious
reconnection along current sheets is absent in the ideal case. If effects of
electron inertia are added, reconnection of magnetic field lines is allowed,
although the resulting model still possesses a noncanonical Hamiltonian
structure. After reviewing the conservation laws of the model equations, the
adopted variational principle with the related conservation laws are described
both at the continuous and discrete level. We verify the favourable properties
of the variational integrator in particular with respect to the preservation of
the invariants of the models under consideration and compare with results from
the literature and those of a pseudo-spectral code.Comment: 35 page
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
An Hp-Adaptive Finite Element Procedure For Fluid-Structure Interaction In Fully Eulerian Framework
This thesis attempts to implement a fully automatic hp-adaptive finite element procedure for fluid-structure interaction (FSI) problems in two dimensions. This work hypotesizes the efficacy of Fully Eulerian framework of FSI in hp-adaptivity on an a posteriori error estimator and adaptation for minimization of error in energy norm. Automatic mesh adaptation over triangular elements is handled by red-green-blue (RGB) refinement method. An effective mesh adaptivity to avoid excessive growth of unknowns is also addressed. Since the hp-method uses high order polynomials as approximation functions, the resulting system matrices are less sparse leading to the notion of FSI computation with parallelism. The parallel hp-adaptive computation is assessed with the conventional uniform and h refinement on a number of benchmark test cases. Subsequently, the efficacy of the fully Eulerian framework is compared to the well known Arbitrary Lagrangian Framework( ALE) for two different material models, namely, the St. Venant Kirchoff and the Neo-Hookean models. It was found that the fully Eulerian framework provides accurate FSI predictions for large deformation without need of frequent remeshing. The hp-adaptive method was also found to be a viable approach in obtaining accurate solutions without much compromise in computer memory and time. Furthermore, the integration of parallelism is successful in reducing the computation time by up to two orders of magnitude relative to the serial solver. For the comparisons between the ALE and the fully Eulerian frameworks, the computed solutions in all test cases are observed to be in agreement with each other
Fluxon Modeling of Low-Beta Plasmas
We have developed a new, quasi-Lagrangian approach for numerical modeling of
magnetohydrodynamics in low to moderate plasmas such as the solar
corona. We introduce the concept of a ``fluxon'', a discretized field line.
Fluxon models represent the magnetic field as a skeleton of such discrete field
lines, and interpolate field values from the geometry of the skeleton where
needed, reversing the usual direction of the field line transform. The fluxon
skeleton forms the grid for a collection of 1-D Eulerian models of plasma along
individual flux tubes. Fluxon models have no numerical resistivity, because
they preserve topology explicitly. Our prototype code, \emph{FLUX}, is
currently able to find 3-D nonlinear force-free field solutions with a
specified field topology, and work is ongoing to validate and extend the code
to full magnetohydrodynamics. FLUX has significant scaling advantages over
conventional models: for ``magnetic carpet'' models, with photospheric
line-tied boundary conditions, FLUX simulations scale in complexity like a
conventional 2-D grid although the full 3-D field is represented. The code is
free software and is available online. In this current paper we introduce
fluxons and our prototype code, and describe the course of future work with the
code.Comment: 14 pages, 11 figures; also in press for JAST
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