Hybridized discontinuous Galerkin methods for magnetohydrodynamics

Discontinuous Galerkin (DG) methods combine the advantages of classical finite element and finite volume methods. Like finite volume methods, through the use of discontinuous spaces in the discrete functional setting, we automatically have local conservation, an essential property for a numerical method to behave well when applied to hyperbolic conservation laws. Like classical finite element methods, DG methods allow for higher order approximations with compact stencils. For time-dependent problems with implicit time stepping and for steady-state problems, DG methods give a larger globally coupled linear system than continuous Galerkin methods (especially for three dimensional problems and low polynomial orders). The primary motivation of the hybridized (or hybridizable) discontinuous Galerkin (HDG) methods is to reduce the number of globally coupled unknowns in DG methods when implicit time stepping or direct-to-steady-state solutions are desired. This is accomplished by the introduction of new “trace unknowns” defined on the mesh skeleton, the definition of one-sided numerical fluxes, and the enforcement of local conservation. This results in a globally coupled linear system where the local “volume unknowns” can be eliminated in a Schur complement procedure, resulting in a reduced globally coupled system in terms of only the trace unknowns. Magnetohydrodynamics (MHD) is the study of the flow of electrically conducting fluids under the influence of magnetic fields. The MHD equations are used to describe important physical phenomena including laboratory plasmas (plasma confinement in fusion energy devices), astrophysical plasmas (solar coronas, planetary magnetospheres) and liquid metal flows (metallurgy processes, the Earth’s molten core, cooling for nuclear reactors). Incompressible MHD, which is the main focus of this work, is relevant in low Lundquist number liquid metals, in high Lundquist number, large guide field fusion plasmas, and in low Mach number compressible flows. The equations of MHD are highly nonlinear, and are characterized by physical phenomena spanning wide ranges of length and time scales. For numerical methods, this presents challenges in both spatial and temporal discretization. In terms of temporal discretization, fully implicit numerical methods are attractive in their robustness; they allow for stable, high-order time integration over long time scales of interest.Computational Science, Engineering, and Mathematic

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 $\mathbf{B}$-$\mathbf{E}$ 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 $S=1{,}000$ by choosing the Rayleigh number in the range between 0 and $100{,}000$ 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

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

A Divergence-Free and H(div)H(div)-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations

We proposed a divergence-free and $H(div)$-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 $H(div)$-conforming for both linear and nonlinear cases. Moreover, the results of well-posedness analysis, divergence-free property, and $H(div)$-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 $L^2$ 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

Fast and scalable solvers for high-order hybridized discontinuous Galerkin methods with applications to fluid dynamics and magnetohydrodynamics

The hybridized discontinuous Galerkin methods (HDG) introduced a decade ago is a promising candidate for high-order spatial discretization combined with implicit/implicit-explicit time stepping. Roughly speaking, HDG methods combines the advantages of both discontinuous Galerkin (DG) methods and hybridized methods. In particular, it enjoys the benefits of equal order spaces, upwinding and ability to handle large gradients of DG methods as well as the smaller globally coupled linear system, adaptivity, and multinumeric capabilities of hybridized methods. However, the main bottleneck in HDG methods, limiting its use to small to moderate sized problems, is the lack of scalable linear solvers. In this thesis we develop fast and scalable solvers for HDG methods consisting of domain decomposition, multigrid and multilevel solvers/preconditioners with an ultimate focus on simulating large scale problems in fluid dynamics and magnetohydrodynamics (MHD). First, we propose a domain decomposition based solver namely iterative HDG for partial differential equations (PDEs). It is a fixed point iterative scheme, with each iteration consisting only of element-by-element and face-by-face embarrassingly parallel solves. Using energy analysis we prove the convergence of the schemes for scalar and system of hyperbolic PDEs and verify the results numerically. We then propose a novel geometric multigrid approach for HDG methods based on fine scale Dirichlet-to-Neumann maps. The algorithm combines the robustness of algebraic multigrid methods due to operator dependent intergrid transfer operators and at the same time has fixed coarse grid construction costs due to its geometric nature. For diffusion dominated PDEs such as the Poisson and the Stokes equations the algorithm gives almost perfect hp--scalability. Next, we propose a multilevel algorithm by combining the concepts of nested dissection, a fill-in reducing ordering strategy, variational structure and high-order properties of HDG, and domain decomposition. Thanks to its root in direct solver strategy the performance of the solver is almost independent of the nature of the PDEs and mostly depends on the smoothness of the solution. We demonstrate this numerically with several prototypical PDEs. Finally, we propose a block preconditioning strategy for HDG applied to incompressible visco-resistive MHD. We use a least squares commutator approximation for the inverse of the Schur complement and algebraic multigrid or the multilevel preconditioner for the approximate inverse of the nodal block. With several 2D and 3D transient examples we demonstrate the robustness and parallel scalability of the block preconditionerAerospace Engineerin

Geometric, Variational Discretization of Continuum Theories

This study derives geometric, variational discretizations of continuum theories arising in fluid dynamics, magnetohydrodynamics (MHD), and the dynamics of complex fluids. A central role in these discretizations is played by the geometric formulation of fluid dynamics, which views solutions to the governing equations for perfect fluid flow as geodesics on the group of volume-preserving diffeomorphisms of the fluid domain. Inspired by this framework, we construct a finite-dimensional approximation to the diffeomorphism group and its Lie algebra, thereby permitting a variational temporal discretization of geodesics on the spatially discretized diffeomorphism group. The extension to MHD and complex fluid flow is then made through an appeal to the theory of Euler-Poincar\'{e} systems with advection, which provides a generalization of the variational formulation of ideal fluid flow to fluids with one or more advected parameters. Upon deriving a family of structured integrators for these systems, we test their performance via a numerical implementation of the update schemes on a cartesian grid. Among the hallmarks of these new numerical methods are exact preservation of momenta arising from symmetries, automatic satisfaction of solenoidal constraints on vector fields, good long-term energy behavior, robustness with respect to the spatial and temporal resolution of the discretization, and applicability to irregular meshes

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