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

Efficient Numerical Methods for Magnetohydrodynamic Flow

This dissertation studies efficient numerical methods for approximating solu-tions to viscous, incompressible, time-dependent magnetohydrodynamic (MHD) flows and computing MHD flows ensembles. Chapter 3 presents and analyzes a fully discrete, decoupled efficient algorithm for MHD flow that is based on the Els¨asser variable formulation, proves its uncondi-tional stability with respect to the timestep size, and proves its unconditional con-vergence. Numerical experiments are given which verify all predicted convergence rates of our analysis, show the results of the scheme on a set of channel flow problems match well the results found when the computation is done with MHD in primitive variables, and finally illustrate that the scheme performs well for channel flow over a step. In chapter 4, we propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep, yet maintains un-conditional stability with respect to timestep size. The scheme is optimally accu-rate in space, and behaves like second order in time in practice. The proposed method chooses θ ∈ [0, 1], dependent on the viscosity ν and magnetic diffusiv-ity νm, so that unconditionally stability is achieved, and gives temporal accuracy O(∆t2 + (1 − θ)|ν − νm|∆t). In practice, ν and νm are small, and so the method be-haves like second order. We show the θ-method provides excellent accuracy in cases where usual BDF2 is unstable. Chapter 5 proposes an efficient algorithm and studies for computing flow en-sembles of incompressible MHD flows under uncertainties in initial or boundary data. The ensemble average of J realizations is approximated through an efficient algo-rithm that, at each time step, uses the same coefficient matrix for each of the J system solves. Hence, preconditioners need to be built only once per time step, and the algorithm can take advantage of block linear solvers. Additionally, an Els¨asser variable formulation is used, which allows for a stable decoupling of each MHD system at each time step. We prove stability and convergence of the algorithm, and test it with two numerical experiments. This work concludes with chapter 6, which proposes, analyzes and tests high order algebraic splitting methods for MHD flows. The key idea is to applying Yosida-type algebraic splitting to the incremental part of the unknowns at each time step. This reduces the block Schur complement by decoupling it into two Navier-Stokes-type Schur complements, each of which is symmetric positive definite and the same at each time step. We prove the splitting is third order in ∆t, and if used together with (block-)pressure correction, is fourth order. A full analysis of the solver is given, both as a linear algebraic approximation, and as a finite element discretization of an approximation to the un-split discrete system. Numerical tests are given to illustrate the theory and show the effectiveness of the method. Finally, conclusions and future works are discussed in the final chapter

A semi-implicit hybrid finite volume / finite element scheme for all Mach number flows on staggered unstructured meshes

In this paper a new hybrid semi-implicit finite volume / finite element (FV/FE) scheme is presented for the numerical solution of the compressible Euler and Navier-Stokes equations at all Mach numbers on unstructured staggered meshes in two and three space dimensions. The chosen grid arrangement consists of a primal simplex mesh composed of triangles or tetrahedra, and an edge-based / face-based staggered dual mesh. The governing equations are discretized in conservation form. The nonlinear convective terms of the equations, as well as the viscous stress tensor and the heat flux, are discretized on the dual mesh at the aid of an explicit local ADER finite volume scheme, while the implicit pressure terms are discretized at the aid of a continuous $\mathbb{P}^{1}$ finite element method on the nodes of the primal mesh. In the zero Mach number limit, the new scheme automatically reduces to the hybrid FV/FE approach forwarded in \cite{BFTVC17} for the incompressible Navier-Stokes equations. As such, the method is asymptotically consistent with the incompressible limit of the governing equations and can therefore be applied to flows at all Mach numbers. Due to the chosen semi-implicit discretization, the CFL restriction on the time step is only based on the magnitude of the flow velocity and not on the sound speed, hence the method is computationally efficient at low Mach numbers. In the chosen discretization, the only unknown is the scalar pressure field at the new time step. Furthermore, the resulting pressure system is symmetric and positive definite and can therefore be very efficiently solved with a matrix-free conjugate gradient method. In order to assess the capabilities of the new scheme, we show computational results for a large set of benchmark problems that range from the quasi incompressible low Mach number regime to compressible flows with shock waves

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

HIGHER ACCURACY METHODS FOR FLUID FLOWS IN VARIOUS APPLICATIONS: THEORY AND IMPLEMENTATION

This dissertation contains research on several topics related to Defect-deferred correction (DDC) method applying to CFD problems. First, we want to improve the error due to temporal discretization for the problem of two convection dominated convection-diffusion problems, coupled across a joint interface. This serves as a step towards investigating an atmosphere-ocean coupling problem with the interface condition that allows for the exchange of energies between the domains. The main diffuculty is to decouple the problem in an unconditionally stable way for using legacy code for subdomains. To overcome the issue, we apply the Deferred Correction (DC) method. The DC method computes two successive approximations and we will exploit this extra flexibility by also introducing the artificial viscosity to resolve the low viscosity issue. The low viscosity issue is to lose an accuracy and a way of finding a approximate solution as a diffusion coeffiscient gets low. Even though that reduces the accuracy of the first approximation, we recover the second order accuracy in the correction step. Overall, we construct a defect and deferred correction (DDC) method. So that not only the second order accuracy in time and space is obtained but the method is also applicable to flows with low viscosity. Upon successfully completing the project in Chapter 1, we move on to implementing similar ideas for a fluid-fluid interaction problem with nonlinear interface condition; the results of this endeavor are reported in Chapter 2. In the third chapter, we represent a way of using an algorithm of an existing penalty-projection for MagnetoHydroDynamics, which allows for the usage of the less sophisticated and more computationally attractive Taylor-Hood pair of finite element spaces. We numerically show that the new modification of the method allows to get first order accuracy in time on the Taylor-Hood finite elements while the existing method would fail on it. In the fourth chapter, we apply the DC method to the magnetohydrodynamic (MHD) system written in Elsásser variables to get second order accuracy in time. We propose and analyze an algorithm based on the penalty projection with graddiv stabilized Taylor Hood solutions of Elsásser formulations

HIGH ACCURACY METHODS AND REGULARIZATION TECHNIQUES FOR FLUID FLOWS AND FLUID-FLUID INTERACTION

This dissertation contains several approaches to resolve irregularity issues of CFD problems, including a decoupling of non-linearly coupled fluid-fluid interaction, due to high Reynolds number. New models present not only regularize the linear systems but also produce high accurate solutions both in space and time. To achieve this goal, methods solve a computationally attractive artificial viscosity approximation of the target problem, and then utilize a correction approach to make it high order accurate. This way, they all allow the usage of legacy code | a frequent requirement in the simulation of fluid flows in complex geometries. In addition, they all pave the way for parallelization of the correction step, which roughly halves the computational time for each method, i.e. solves at about the same time that is required for DNS with artificial viscosity. Also, methods present do not requires all over function evaluations as one can store them, and reuse for the correction steps. All of the chapters in this dissertation are self-contained, and introduce model first, and then present both theoretical and computational findings of the corresponding method

Numerical methods for large-scale, time-dependent partial differential equations

A survey of numerical methods for time dependent partial differential equations is presented. The emphasis is on practical applications to large scale problems. A discussion of new developments in high order methods and moving grids is given. The importance of boundary conditions is stressed for both internal and external flows. A description of implicit methods is presented including generalizations to multidimensions. Shocks, aerodynamics, meteorology, plasma physics and combustion applications are also briefly described

Research in progress and other activities of the Institute for Computer Applications in Science and Engineering

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics and computer science during the period April 1, 1993 through September 30, 1993. The major categories of the current ICASE research program are: (1) applied and numerical mathematics, including numerical analysis and algorithm development; (2) theoretical and computational research in fluid mechanics in selected areas of interest to LaRC, including acoustic and combustion; (3) experimental research in transition and turbulence and aerodynamics involving LaRC facilities and scientists; and (4) computer science