First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.Comment: 22 pages, 8 figures submitted to Journal of Computational Physic

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 novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics

In this work we introduce a novel semi-implicit structure-preserving finite-volume/finite-difference scheme for the viscous and resistive equations of magnetohydrodynamics (MHD) based on an appropriate 3-split of the governing PDE system, which is decomposed into a first convective subsystem, a second subsystem involving the coupling of the velocity field with the magnetic field and a third subsystem involving the pressure-velocity coupling. The nonlinear convective terms are discretized explicitly, while the remaining two subsystems accounting for the Alfven waves and the magneto-acoustic waves are treated implicitly. The final algorithm is at least formally constrained only by a mild CFL stability condition depending on the velocity field of the pure hydrodynamic convection. To preserve the divergence-free constraint of the magnetic field exactly at the discrete level, a proper set of overlapping dual meshes is employed. The resulting linear algebraic systems are shown to be symmetric and therefore can be solved by means of an efficient standard matrix-free conjugate gradient algorithm. One of the peculiarities of the presented algorithm is that the magnetic field is defined on the edges of the main grid, while the electric field is on the faces. The final scheme can be regarded as a novel shock-capturing, conservative and structure preserving semi-implicit scheme for the nonlinear viscous and resistive MHD equations. Several numerical tests are presented to show the main features of our novel solver: linear-stability in the sense of Lyapunov is verified at a prescribed constant equilibrium solution; a 2nd-order of convergence is numerically estimated; shock-capturing capabilities are proven against a standard set of stringent MHD shock-problems; accuracy and robustness are verified against a nontrivial set of 2- and 3-dimensional MHD problems.Comment: 44 pages, 22 figures, 2 table

Parallel Efficiency-based Adaptive Local Refinement

New adaptive local refinement (ALR) strategies are developed, the goal of which is to reach a given error tolerance with the least amount of computational cost. This strategy is especially attractive in the setting of a first-order system least-squares (FOSLS) finite element formulation in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). To accomplish this, the refinement decisions are determined based on minimizing the predicted `accuracy-per-computational-cost\u27 efficiency (ACE). The nested iteration approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Efficiency of the algorithms are demonstrated through a 2D Poisson problem with steep gradients, and 2D reduced model of the incompressible, resistive magnetohydrodynamic (MHD) equations. Accommodations of the ALR strategies to parallel computer architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quad-tree structures and a space filling curve. This automatically ameliorates load balancing issues at finer levels. Numerical results produced on Frost, the NCAR/CU Blue Gene/L supercomputer, are presented for a 2D Poisson problem with steep gradients, a 2D backward facing step incompressible Stokes equations and Navier-Stokes equations. The NI-FOSL-AMG-ACE approach is able to provide highly resolved approximations to rapidly varying solutions using a small number of work units. Excellent weak and strong scalability of parallel ALR are demonstrated on up to 4,096 processors for problems with up to 15 million biquadratic elements

On stabilized finite element methods based on the Scott-Zhang projector: circumventing the inf-sup condition for the Stokes problem

In this work we propose a stabilized nite element method that permits us to circumvent discrete inf-sup conditions, e.g. allowing equal order interpolation. The type of method we propose belongs to the family of symmetric stabilization techniques, which are based on the introduction of additional terms that penalize the di erence between some quantities, i.e. the pressure gradient in the Stokes problem, and their nite element projections. The key feature of the formulation we propose is the de nition of the projection to be used, a non-standard Scott-Zhang projector that is well-de ned for L1() functions. The resulting method has some appealing features: the projector is local and nested meshes or enriched spaces are not required

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations. This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical non-dimensional parameters and with respect to mesh refinement

Topics in Magnetohydrodynamics

To understand plasma physics intuitively one need to master the MHD behaviors. As sciences advance, gap between published textbooks and cutting-edge researches gradually develops. Connection from textbook knowledge to up-to-dated research results can often be tough. Review articles can help. This book contains eight topical review papers on MHD. For magnetically confined fusion one can find toroidal MHD theory for tokamaks, magnetic relaxation process in spheromaks, and the formation and stability of field-reversed configuration. In space plasma physics one can get solar spicules and X-ray jets physics, as well as general sub-fluid theory. For numerical methods one can find the implicit numerical methods for resistive MHD and the boundary control formalism. For low temperature plasma physics one can read theory for Newtonian and non-Newtonian fluids etc