103 research outputs found
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
A monolithic approach for the incompressible magnetohydrodynamics equations
A numerical algorithm has been developed to solve the incompressible magnetohydrodynamics (MHD) equations in a fully coupled form. The numerical approach is based on the side
centered finite volume approximation where the velocity and magnetic filed vector components are
defined at the center of edges/faces, meanwhile the pressure term is defined at the element
centroid. In order to enforce a divergence free magnetic field, a magnetic pressure is introduced
to the induction equation. The resulting large-scale algebraic linear equations are solved using a
one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within
each partitioned sub-domains. The parallel implementation of the present fully coupled
unstructured MHD solver is based on the PETSc library for improving the effi- ciency of the
parallel algorithm. The numerical algorithm is validated for 2D lid-driven cavity
flows and backward step problems for both conducting and insulating walls
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
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
page
A local Fourier analysis of additive Vanka relaxation for the Stokes equations
Multigrid methods are popular solution algorithms for many discretized PDEs,
either as standalone iterative solvers or as preconditioners, due to their high
efficiency. However, the choice and optimization of multigrid components such
as relaxation schemes and grid-transfer operators is crucial to the design of
optimally efficient algorithms. It is well--known that local Fourier analysis
(LFA) is a useful tool to predict and analyze the performance of these
components. In this paper, we develop a local Fourier analysis of monolithic
multigrid methods based on additive Vanka relaxation schemes for mixed
finite-element discretizations of the Stokes equations. The analysis offers
insight into the choice of "patches" for the Vanka relaxation, revealing that
smaller patches offer more effective convergence per floating point operation.
Parameters that minimize the two-grid convergence factor are proposed and
numerical experiments are presented to validate the LFA predictions.Comment: 30 pages, 12 figures. Add new sections: multiplicative Vanka results
and sensitivity of convergence factors to mesh distortio
Exploiting mesh structure to improve multigrid performance for saddle point problems
In recent years, solvers for finite-element discretizations of linear or
linearized saddle-point problems, like the Stokes and Oseen equations, have
become well established. There are two main classes of preconditioners for such
systems: those based on block-factorization approach and those based on
monolithic multigrid. Both classes of preconditioners have several critical
choices to be made in their composition, such as the selection of a suitable
relaxation scheme for monolithic multigrid. From existing studies, some insight
can be gained as to what options are preferable in low-performance computing
settings, but there are very few fair comparisons of these approaches in the
literature, particularly for modern architectures, such as GPUs. In this paper,
we perform a comparison between a block-triangular preconditioner and a
monolithic multigrid method with the three most common choices of relaxation
scheme - Braess-Sarazin, Vanka, and Schur-Uzawa. We develop a performant Vanka
relaxation algorithm for structured-grid discretizations, which takes advantage
of memory efficiencies in this setting. We detail the behavior of the various
CUDA kernels for the multigrid relaxation schemes and evaluate their individual
arithmetic intensity, performance, and runtime. Running a preconditioned FGMRES
solver for the Stokes equations with these preconditioners allows us to compare
their efficiency in a practical setting. We show monolithic multigrid can
outperform block-triangular preconditioning, and that using Vanka or
Braess-Sarazin relaxation is most efficient. Even though multigrid with Vanka
relaxation exhibits reduced performance on the CPU (up to slower than
Braess-Sarazin), it is able to outperform Braess-Sarazin by more than on
the GPU, making it a competitive algorithm, especially given the high amount of
algorithmic tuning needed for effective Braess-Sarazin relaxation.Comment: submitted to IJHPC
A Conservative Finite Element Solver for MHD Kinematics equations: Vector Potential method and Constraint Preconditioning
A new conservative finite element solver for the three-dimensional steady
magnetohydrodynamic (MHD) kinematics equations is presented.The solver utilizes
magnetic vector potential and current density as solution variables, which are
discretized by H(curl)-conforming edge-element and H(div)-conforming face
element respectively. As a result, the divergence-free constraints of discrete
current density and magnetic induction are both satisfied. Moreover the
solutions also preserve the total magnetic helicity. The generated linear
algebraic equation is a typical dual saddle-point problem that is
ill-conditioned and indefinite. To efficiently solve it, we develop a block
preconditioner based on constraint preconditioning framework and devise a
preconditioned FGMRES solver. Numerical experiments verify the conservative
properties, the convergence rate of the discrete solutions and the robustness
of the preconditioner.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1712.0892
Monolithic multigrid methods for high-order discretizations of time-dependent PDEs
A currently growing interest is seen in developing solvers that couple high-fidelity and
higher-order spatial discretization schemes with higher-order time stepping methods
for various time-dependent fluid plasma models. These problems are famously known
to be stiff, thus only implicit time-stepping schemes with certain stability properties
can be used. Of the most powerful choices are the implicit Runge-Kutta methods
(IRK). However, they are multi-stage, often producing a very large and nonsymmetric
system of equations that needs to be solved at each time step. There have been recent
efforts on developing efficient and robust solvers for these systems. We have accomplished
this by using a Newton-Krylov-multigrid approach that applies a multigrid
preconditioner monolithically, preserving the system couplings, and uses Newton’s
method for linearization wherever necessary. We show robustness of our solver on the
single-fluid magnetohydrodynamic (MHD) model, along with the (Navier-)Stokes and
Maxwell’s equations. For all these, we couple IRK with higher-order (mixed) finiteelement
(FEM) spatial discretizations. In the Navier-Stokes problem, we further
explore achieving more higher-order approximations by using nonconforming mixed
FEM spaces with added penalty terms for stability. While in the Maxwell problem,
we focus on the rarely used E-B form, where both electric and magnetic fields are
differentiated in time, and overcome the difficulty of using FEM on curved domains
by using an elasticity solve on each level in the non-nested hierarchy of meshes in the
multigrid method
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