A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD

Cyr, Eric C; Shadid, John N.; Tuminaro, Raymond S.; Pawlowski, Roger P.; Chacòn, Luis

doi:10.1137/12088879x

Cited by 66 publications

(64 citation statements)

References 34 publications

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“…For realistic applications where the systems are very large, effective preconditioning of iterative methods is required for efficiency. Some recently developed solvers for fully coupled MHD formulations include a coupled AMG preconditioned Newton-Krylov method for a vector potential formulation [23], a multigrid preconditioned Newton-Krylov method for a parabolic reformulation of the MHD equations [1], and a block preconditioned Newton-Krylov method for a vector potential formulation [4].…”

Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

“…Like (u, p, B), the ordering (u, B, p) gives rise to Schur complements that are nested, multi-term, and for this reason, we will not pursue this ordering further. We note that preconditioning a system similar in structure to that obtained from the (u, p, B) ordering has been studied from another perspective in [4].…”

Section: A Block Preconditionermentioning

confidence: 99%

“…For solves on the pressure space we use one V-cycle of AMG with five sweeps of a Gauss-Seidel smoother. This AMG technology has been demonstrated to be algorithmically scalable for both an equal order stabilized finite element formulation of the full MHD system and as a component solve in physics based preconditioners [4,23]. All problems were run on the Red Sky computer at Sandia National Laboratories.…”

Section: A Blockmentioning

confidence: 99%

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A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

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Abstract. 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. Key words. magnetohydrodynamics, iterative methods, preconditionersAMS subject classifications. 76W05, 65F08, 65M221. Introduction. The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics [13]. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis [19]. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. Because multiple physical processes are represented in the model, the PDEs can span over a range of length-and time-scales, making the equations difficult to solve and requiring a robust, accurate means of approximating the solution. Decoupled solution methods which solve the fluid and magnetic systems separately and possibly couple the systems by an outer iteration have been commonly employed as solvers for the transient and steady MHD systems. In the context of transient systems these methods are commonly used in operator splitting techniques, for steady state solves a fixed point iteration serves to couple the system (see e.g. [2], and th...

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Section: Phillips Elman Cyr Shadid and Pawlowskimentioning

confidence: 99%

Section: A Block Preconditionermentioning

confidence: 99%

Section: A Blockmentioning

confidence: 99%

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A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Phillips¹,

Elman²,

Cyr³

et al. 2014

SIAM J. Sci. Comput.

Self Cite

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“…They investigate the performance of one-level Schwarz method and also a new fully coupled algebraic multilevel method in that paper. In [30,9,23], they explore a class of robust and scalable parallel preconditioners for Newton-Krylov solver based on the physical-based approximate block factorization (ABF) technique. They employ block factorization and approximate the resulting Schur complement recursively based on special techniques, for example, operator commutativity [12].…”

Section: Introductionmentioning

confidence: 99%

Robust preconditioners for incompressible MHD models

et al. 2016

Journal of Computational Physics

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ABSTRACT. In this paper, we develop two classes of robust preconditioners for the structurepreserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.

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“…This type of preconditioners have been widely studied for the incompressible Navier-Stokes equations in the fluid mechanics community [22,21,18]. Recently, this approach has been applied to the full resistive MHD equations [9] (using the preconditioner as a solver in an operator splitting fashion) and to the 2D incompressible (reduced) resistive MHD formulation [19]. The crucial aspect in these approximate block preconditioners relies in an efficient approximation of the Schur complement that allows the uncoupling between the several physical variables of the problem at the preconditioner level.…”

Section: Introductionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential 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 different 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 efficient 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 flexible and general software design for the code implementation of this type of preconditioners.

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A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD

Cited by 66 publications

References 34 publications

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Robust preconditioners for incompressible MHD models

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

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