A fully implicit Newton-Krylov-Schwarz method for tokamak magnetohydrodynamics: Jacobian construction and preconditioner formulation

Handle URI:
http://hdl.handle.net/10754/562052
Title:
A fully implicit Newton-Krylov-Schwarz method for tokamak magnetohydrodynamics: Jacobian construction and preconditioner formulation
Authors:
Reynolds, Daniel R.; Samtaney, Ravi ( 0000-0002-4702-6473 ) ; Tiedeman, Hilari C.
Abstract:
Single-fluid resistive magnetohydrodynamics (MHD) is a fluid description of fusion plasmas which is often used to investigate macroscopic instabilities in tokamaks. In MHD modeling of tokamaks, it is often desirable to compute MHD phenomena to resistive time scales or a combination of resistive-Alfvén time scales, which can render explicit time stepping schemes computationally expensive. We present recent advancements in the development of preconditioners for fully nonlinearly implicit simulations of single-fluid resistive tokamak MHD. Our work focuses on simulations using a structured mesh mapped into a toroidal geometry with a shaped poloidal cross-section, and a finite-volume spatial discretization of the partial differential equation model. We discretize the temporal dimension using a fully implicit or the backwards differentiation formula method, and solve the resulting nonlinear algebraic system using a standard inexact Newton-Krylov approach, provided by the sundials library. The focus of this paper is on the construction and performance of various preconditioning approaches for accelerating the convergence of the iterative solver algorithms. Effective preconditioners require information about the Jacobian entries; however, analytical formulae for these Jacobian entries may be prohibitive to derive/implement without error. We therefore compute these entries using automatic differentiation with OpenAD. We then investigate a variety of preconditioning formulations inspired by standard solution approaches in modern MHD codes, in order to investigate their utility in a preconditioning context. We first describe the code modifications necessary for the use of the OpenAD tool and sundials solver library. We conclude with numerical results for each of our preconditioning approaches in the context of pellet-injection fueling of tokamak plasmas. Of these, our optimal approach results in a speedup of a factor of 3 compared with non-preconditioned implicit tests, with that performance gap rapidly widening with increasing mesh refinement. © 2012 IOP Publishing Ltd.
KAUST Department:
Mechanical Engineering Program; Physical Sciences and Engineering (PSE) Division; Clean Combustion Research Center; Fluid and Plasma Simulation Group (FPS)
Publisher:
IOP Publishing
Journal:
Computational Science & Discovery
Issue Date:
1-Jan-2012
DOI:
10.1088/1749-4699/5/1/014003
Type:
Article
ISSN:
17494680
Appears in Collections:
Articles; Physical Sciences and Engineering (PSE) Division; Mechanical Engineering Program; Clean Combustion Research Center

Full metadata record

DC FieldValue Language
dc.contributor.authorReynolds, Daniel R.en
dc.contributor.authorSamtaney, Ravien
dc.contributor.authorTiedeman, Hilari C.en
dc.date.accessioned2015-08-03T09:43:38Zen
dc.date.available2015-08-03T09:43:38Zen
dc.date.issued2012-01-01en
dc.identifier.issn17494680en
dc.identifier.doi10.1088/1749-4699/5/1/014003en
dc.identifier.urihttp://hdl.handle.net/10754/562052en
dc.description.abstractSingle-fluid resistive magnetohydrodynamics (MHD) is a fluid description of fusion plasmas which is often used to investigate macroscopic instabilities in tokamaks. In MHD modeling of tokamaks, it is often desirable to compute MHD phenomena to resistive time scales or a combination of resistive-Alfvén time scales, which can render explicit time stepping schemes computationally expensive. We present recent advancements in the development of preconditioners for fully nonlinearly implicit simulations of single-fluid resistive tokamak MHD. Our work focuses on simulations using a structured mesh mapped into a toroidal geometry with a shaped poloidal cross-section, and a finite-volume spatial discretization of the partial differential equation model. We discretize the temporal dimension using a fully implicit or the backwards differentiation formula method, and solve the resulting nonlinear algebraic system using a standard inexact Newton-Krylov approach, provided by the sundials library. The focus of this paper is on the construction and performance of various preconditioning approaches for accelerating the convergence of the iterative solver algorithms. Effective preconditioners require information about the Jacobian entries; however, analytical formulae for these Jacobian entries may be prohibitive to derive/implement without error. We therefore compute these entries using automatic differentiation with OpenAD. We then investigate a variety of preconditioning formulations inspired by standard solution approaches in modern MHD codes, in order to investigate their utility in a preconditioning context. We first describe the code modifications necessary for the use of the OpenAD tool and sundials solver library. We conclude with numerical results for each of our preconditioning approaches in the context of pellet-injection fueling of tokamak plasmas. Of these, our optimal approach results in a speedup of a factor of 3 compared with non-preconditioned implicit tests, with that performance gap rapidly widening with increasing mesh refinement. © 2012 IOP Publishing Ltd.en
dc.publisherIOP Publishingen
dc.titleA fully implicit Newton-Krylov-Schwarz method for tokamak magnetohydrodynamics: Jacobian construction and preconditioner formulationen
dc.typeArticleen
dc.contributor.departmentMechanical Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentClean Combustion Research Centeren
dc.contributor.departmentFluid and Plasma Simulation Group (FPS)en
dc.identifier.journalComputational Science & Discoveryen
dc.contributor.institutionDepartment of Mathematics, Southern Methodist University, Dallas, TX 75275-0156, United Statesen
kaust.authorSamtaney, Ravien
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.