288 research outputs found
Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code
In this paper we present a rigorous derivation of the reduced MHD models with
and without parallel velocity that are implemented in the non-linear MHD code
JOREK. The model we obtain contains some terms that have been neglected in the
implementation but might be relevant in the non-linear phase. These are
necessary to guarantee exact conservation with respect to the full MHD energy.
For the second part of this work, we have replaced the linearized time stepping
of JOREK by a non-linear solver based on the Inexact Newton method including
adaptive time stepping. We demonstrate that this approach is more robust
especially with respect to numerical errors in the saturation phase of an
instability and allows to use larger time steps in the non-linear phase
Monolithic Multigrid for Magnetohydrodynamics
The magnetohydrodynamics (MHD) equations model a wide range of plasma physics
applications and are characterized by a nonlinear system of partial
differential equations that strongly couples a charged fluid with the evolution
of electromagnetic fields. After discretization and linearization, the
resulting system of equations is generally difficult to solve due to the
coupling between variables, and the heterogeneous coefficients induced by the
linearization process. In this paper, we investigate multigrid preconditioners
for this system based on specialized relaxation schemes that properly address
the system structure and coupling. Three extensions of Vanka relaxation are
proposed and applied to problems with up to 170 million degrees of freedom and
fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to
20,000 for time-dependent problems
Space-Time Block Preconditioning for Incompressible Resistive Magnetohydrodynamics
This work develops a novel all-at-once space-time preconditioning approach
for resistive magnetohydrodynamics (MHD), with a focus on model problems
targeting fusion reactor design. We consider parallel-in-time due to the long
time domains required to capture the physics of interest, as well as the
complexity of the underlying system and thereby computational cost of long-time
integration. To ameliorate this cost by using many processors, we thus develop
a novel approach to solving the whole space-time system that is parallelizable
in both space and time. We develop a space-time block preconditioning for
resistive MHD, following the space-time block preconditioning concept first
introduced by Danieli et al. in 2022 for incompressible flow, where an
effective preconditioner for classic sequential time-stepping is extended to
the space-time setting. The starting point for our derivation is the continuous
Schur complement preconditioner by Cyr et al. in 2021, which we proceed to
generalise in order to produce, to our knowledge, the first space-time block
preconditioning approach for the challenging equations governing incompressible
resistive MHD. The numerical results are promising for the model problems of
island coalescence and tearing mode, with the overhead computational cost
associated with space-time preconditioning versus sequential time-stepping
being modest and primarily in the range of 2x-5x, which is low for
parallel-in-time schemes in general. Additionally, the scaling results for
inner (linear) and outer (nonlinear) iterations are flat in the case of fixed
time-step size and only grow very slowly in the case of time-step refinement.Comment: 25 pages, 4 figures, 3 table
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
Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem
The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the ow of an electrically charged fuid under the in uence of an external electromagnetic eld with thermal coupling. This system of partial di erential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the di erent physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task
which requires e cient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure,
current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one
of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a fexible and general software design for the code implementation of this type of preconditioners.Preprin
Discretisations and Preconditioners for Magnetohydrodynamics Models
The magnetohydrodynamics (MHD) equations are generally known to be difficult
to solve numerically, due to their highly nonlinear structure and the strong
coupling between the electromagnetic and hydrodynamic variables, especially for
high Reynolds and coupling numbers.
In the first part of this work, we present a scalable augmented Lagrangian
preconditioner for a finite element discretisation of the
- formulation of the incompressible viscoresistive MHD
equations. For stationary problems, our solver achieves robust performance with
respect to the Reynolds and coupling numbers in two dimensions and good results
in three dimensions. Our approach relies on specialised parameter-robust
multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme
ensures exactly divergence-free approximations of both the velocity and the
magnetic field up to solver tolerances.
In the second part, we focus on incompressible, resistive Hall MHD models and
derive structure-preserving finite element methods for these equations. We
present a variational formulation of Hall MHD that enforces the magnetic
Gauss's law precisely (up to solver tolerances) and prove the well-posedness of
a Picard linearisation. For the transient problem, we present time
discretisations that preserve the energy and magnetic and hybrid helicity
precisely in the ideal limit for two types of boundary conditions.
In the third part, we investigate anisothermal MHD models. We start by
performing a bifurcation analysis for a magnetic Rayleigh--B\'enard problem at
a high coupling number by choosing the Rayleigh number in the range
between 0 and as the bifurcation parameter. We study the effect of
the coupling number on the bifurcation diagram and outline how we create
initial guesses to obtain complex solution patterns and disconnected branches
for high coupling numbers.Comment: Doctoral thesis, Mathematical Institute, University of Oxford. 174
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