357 research outputs found
Modeling anisotropic diffusion using a departure from isotropy approach
There are a large number of finite volume solvers available for solution of isotropic diffusion equation. This article presents an approach of adapting these solvers to solve anisotropic diffusion equations. The formulation works by decomposing the diffusive flux into a component associated with isotropic diffusion and another component associated with departure from isotropic diffusion. This results in an isotropic diffusion equation with additional terms to account for the anisotropic effect. These additional terms are treated using a deferred correction approach and coupled via an iterative procedure. The presented approach is validated against various diffusion problems in anisotropic media with known analytical or numerical solutions. Although demonstrated for two-dimensional problems, extension of the present approach to three-dimensional problems is straight forward. Other than the finite volume method, this approach can be applied to any discretization method
High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs
Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching
Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids
We present and discuss three discontinuous Galerkin (dG) discretizations for
the anisotropic heat conduction equation on non-aligned cylindrical grids. Our
most favourable scheme relies on a self-adjoint local dG (LDG) discretization
of the elliptic operator. It conserves the energy exactly and converges with
arbitrary order. The pollution by numerical perpendicular heat fluxes degrades
with superconvergence rates. We compare this scheme with aligned schemes that
are based on the flux-coordinate independent approach for the discretization of
parallel derivatives. Here, the dG method provides the necessary interpolation.
The first aligned discretization can be used in an explicit time-integrator.
However, the scheme violates conservation of energy and shows up stagnating
convergence rates for very high resolutions. We overcome this partly by using
the adjoint of the parallel derivative operator to construct a second
self-adjoint aligned scheme. This scheme preserves energy, but reveals
unphysical oscillations in the numerical tests, which result in a decreased
order of convergence. Both aligned schemes exhibit low numerical heat fluxes
into the perpendicular direction. We build our argumentation on various
numerical experiments on all three schemes for a general axisymmetric magnetic
field, which is closed by a comparison to the aligned finite difference (FD)
schemes of References [1,2
Finite-difference schemes for anisotropic diffusion
In fusion plasmas diffusion tensors are extremely anisotropic due to the high temperature and large magnetic field strength. This causes diffusion, heat conduction, and viscous momentum loss, to effectively be aligned with the magnetic field lines. This alignment leads to different values for the respective diffusive coefficients in the magnetic field direction and in the perpendicular direction, to the extent that heat diffusion coefficients can be up to 10 to the 12 th times larger in the parallel direction than in the perpendicular direction. This anisotropy puts stringent requirements on the numerical methods used to approximate the MHD-equations since any misalignment of the grid may cause the perpendicular diffusion to be polluted by the numerical error in approximating the parallel diffusion. Currently the common approach is to apply magnetic field-aligned coordinates, an approach that automatically takes care of the directionality of the diffusive coefficients. This approach runs into problems at x-points and at points where there is magnetic re-connection, since this causes local non-alignment. It is therefore useful to consider numerical schemes that are tolerant to the misalignment of the grid with the magnetic field lines, both to improve existing methods and to help open the possibility of applying regular non-aligned grids. To investigate this, in this paper several discretisation schemes are developed and applied to the anisotropic heat diffusion equation on a non-aligned grid.</p
A mimetic finite difference based quasi-static magnetohydrodynamic solver for force-free plasmas in tokamak disruptions
Force-free plasmas are a good approximation where the plasma pressure is tiny
compared with the magnetic pressure, which is the case during the cold vertical
displacement event (VDE) of a major disruption in a tokamak. On time scales
long compared with the transit time of Alfven waves, the evolution of a
force-free plasma is most efficiently described by the quasi-static
magnetohydrodynamic (MHD) model, which ignores the plasma inertia. Here we
consider a regularized quasi-static MHD model for force-free plasmas in tokamak
disruptions and propose a mimetic finite difference (MFD) algorithm. The full
geometry of an ITER-like tokamak reactor is treated, with a blanket module
region, a vacuum vessel region, and the plasma region. Specifically, we develop
a parallel, fully implicit, and scalable MFD solver based on PETSc and its
DMStag data structure for the discretization of the five-field quasi-static
perpendicular plasma dynamics model on a 3D structured mesh. The MFD spatial
discretization is coupled with a fully implicit DIRK scheme. The algorithm
exactly preserves the divergence-free condition of the magnetic field under the
resistive Ohm's law. The preconditioner employed is a four-level fieldsplit
preconditioner, which is created by combining separate preconditioners for
individual fields, that calls multigrid or direct solvers for sub-blocks or
exact factorization on the separate fields. The numerical results confirm the
divergence-free constraint is strongly satisfied and demonstrate the
performance of the fieldsplit preconditioner and overall algorithm. The
simulation of ITER VDE cases over the actual plasma current diffusion time is
also presented.Comment: 43 page
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