53 research outputs found
A Divergence-Free and -Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations
We proposed a divergence-free and -conforming embedded-hybridized
discontinuous Galerkin (E-HDG) method for solving stationary incompressible
viso-resistive magnetohydrodynamic (MHD) equations. In particular, the E-HDG
method is computationally far more advantageous over the hybridized
discontinuous Galerkin (HDG) counterpart in general. The benefit is even
significant in the three-dimensional/high-order/fine mesh scenario. On a
simplicial mesh, our method with a specific choice of the approximation spaces
is proved to be well-posed for the linear case. Additionally, the velocity and
magnetic fields are divergence-free and -conforming for both linear and
nonlinear cases. Moreover, the results of well-posedness analysis,
divergence-free property, and -conformity can be directly applied to
the HDG version of the proposed approach. The HDG or E-HDG method for the
linearized MHD equations can be incorporated into the fixed point Picard
iteration to solve the nonlinear MHD equations in an iterative manner. We
examine the accuracy and convergence of our E-HDG method for both linear and
nonlinear cases through various numerical experiments including two- and
three-dimensional problems with smooth and singular solutions. For smooth
problems, the results indicate that convergence rates in the norm for the
velocity and magnetic fields are optimal in the regime of low Reynolds number
and magnetic Reynolds number. Furthermore, the divergence error is machine zero
for both smooth and singular problems. Finally, we numerically demonstrated
that our proposed method is pressure-robust
High order direct Arbitrary-Lagrangian-Eulerian schemes on moving Voronoi meshes with topology changes
We present a new family of very high order accurate direct
Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume (FV) and Discontinuous
Galerkin (DG) schemes for the solution of nonlinear hyperbolic PDE systems on
moving 2D Voronoi meshes that are regenerated at each time step and which
explicitly allow topology changes in time.
The Voronoi tessellations are obtained from a set of generator points that
move with the local fluid velocity. We employ an AREPO-type approach, which
rapidly rebuilds a new high quality mesh rearranging the element shapes and
neighbors in order to guarantee a robust mesh evolution even for vortex flows
and very long simulation times. The old and new Voronoi elements associated to
the same generator are connected to construct closed space--time control
volumes, whose bottom and top faces may be polygons with a different number of
sides. We also incorporate degenerate space--time sliver elements, needed to
fill the space--time holes that arise because of topology changes. The final
ALE FV-DG scheme is obtained by a redesign of the fully discrete direct ALE
schemes of Boscheri and Dumbser, extended here to moving Voronoi meshes and
space--time sliver elements. Our new numerical scheme is based on the
integration over arbitrary shaped closed space--time control volumes combined
with a fully-discrete space--time conservation formulation of the governing PDE
system. In this way the discrete solution is conservative and satisfies the GCL
by construction.
Numerical convergence studies as well as a large set of benchmarks for
hydrodynamics and magnetohydrodynamics (MHD) demonstrate the accuracy and
robustness of the proposed method. Our numerical results clearly show that the
new combination of very high order schemes with regenerated meshes with
topology changes lead to substantial improvements compared to direct ALE
methods on conforming meshes
The Virtual Element Method for the 3D Resistive Magnetohydrodynamic model
We present a four-field Virtual Element discretization for the time-dependent
resistive Magnetohydrodynamics equations in three space dimensions, focusing on
the semi-discrete formulation. The proposed method employs general polyhedral
meshes and guarantees velocity and magnetic fields that are divergence free up
to machine precision. We provide a full convergence analysis under suitable
regularity assumptions, which is validated by some numerical tests
Error Analysis of a Fully Discrete Projection Method for Magnetohydrodynamic System
In this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi-implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme
Robust Finite Elements for linearized Magnetohydrodynamics
We introduce a pressure robust Finite Element Method for the linearized
Magnetohydrodynamics equations in three space dimensions, which is provably
quasi-robust also in the presence of high fluid and magnetic Reynolds numbers.
The proposed scheme uses a non-conforming BDM approach with suitable DG terms
for the fluid part, combined with an -conforming choice for the magnetic
fluxes. The method introduces also a specific CIP-type stabilization associated
to the coupling terms. Finally, the theoretical result are further validated by
numerical experiments
Nonconforming Virtual Element basis functions for space-time Discontinuous Galerkin schemes on unstructured Voronoi meshes
We introduce a new class of Discontinuous Galerkin (DG) methods for solving
nonlinear conservation laws on unstructured Voronoi meshes that use a
nonconforming Virtual Element basis defined within each polygonal control
volume. The basis functions are evaluated as an L2 projection of the virtual
basis which remains unknown, along the lines of the Virtual Element Method
(VEM). Contrarily to the VEM approach, the new basis functions lead to a
nonconforming representation of the solution with discontinuous data across the
element boundaries, as typically employed in DG discretizations. To improve the
condition number of the resulting mass matrix, an orthogonalization of the full
basis is proposed. The discretization in time is carried out following the ADER
(Arbitrary order DERivative Riemann problem) methodology, which yields one-step
fully discrete schemes that make use of a coupled space-time representation of
the numerical solution. The space-time basis functions are constructed as a
tensor product of the virtual basis in space and a one-dimensional Lagrange
nodal basis in time. The resulting space-time stiffness matrix is stabilized by
an extension of the dof-dof stabilization technique adopted in the VEM
framework, hence allowing an element-local space-time Galerkin finite element
predictor to be evaluated. The novel methods are referred to as VEM-DG schemes,
and they are arbitrarily high order accurate in space and time. The new VEM-DG
algorithms are rigorously validated against a series of benchmarks in the
context of compressible Euler and Navier-Stokes equations. Numerical results
are verified with respect to literature reference solutions and compared in
terms of accuracy and computational efficiency to those obtained using a
standard modal DG scheme with Taylor basis functions. An analysis of the
condition number of the mass and space-time stiffness matrix is also forwarded
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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