A fully implicit numerical method for single-fluid resistive magnetohydrodynamics

Implicit Numerical Methods for Magnetohydrodynamics

“…This section is essentially based on the work by Reynolds et al (2006) in which they have developed a fully implicit Jacobian-Free NK method for compressible MHD. The main difference between this section and the previous one is in the preconditioning strategy employed during the Krylov step.…”

“…The initial conditions consist of a perturbed Harris sheet configuration as described in Birn & et al (2001a). Reynolds et al (2006) computed the GEM reconnection challenge problem with a Newton-Krylov method without preconditioning and reproduced the expected Sweet-Parker scaling for the reconnection rate for Lundquist numbers ranging from S = 200 − 10 4 . Furthermore, for a mesh resolution of 512 × 256 their implicit method (without preconditioning) achieved a speedup of about 5.6 compared with an explicit method.…”

“…Chacón (2004) tested the evolution of a magnetosonic wave to verify that the method had low dissipation using a Newton-Krylov approach but without any preconditioning. Reynolds et al (2006) also tested the evolution of a slow magnetosonic wave propagating 45 deg to the mesh, with a Newton-Krylov solver without preconditioning. Numerical tests at 256 2 mesh resolution in 2D indicated that even without preconditioning, the implicit NK method yielded over a factor of ten decrease in CPU time.…”

The Jacobi Spectral Method for Nonlinear Magneto- Hydrodynamic Jeffery-Hamel Problem

“…Therefore, some numerical techniques have been used to obtain the approximate solutions for the MHD flow problems. For more studies on MHD, the interested reader is referred to Reynolds et al (2006); Hughes and Young (1966); Hughes and McNab (1983); Hughes et al (1995); Kao et al (2009);Ni et al (2007); Pericleous and Bojarevies (2007); Pericleous et al (1994); Ramos and Winowich (1990); Slone et al (2003); Takhar et al (2002); Yee and Sjgreen (2007). One type of MHD problems is Jeffery-Hamel problem which is related to the study of laminar boundary layers exhibiting similarity in fluid mechanics.…”

Additive Schwarz-based fully coupled implicit methods for resistive Hall magnetohydrodynamic problems

“…Excellent results with large time steps are reported based on single processor calculations. In [26], a fully parallel, conservative, nonlinearly implicit numerical method is proposed for the integration of the single-fluid resistive MHD system of equations, where a variant of the matrix-free Newton-Krylov method, without preconditioning, is used in conjunction with an adaptive time integration scheme and variable spatial discretization accuracy. In this approach, the time step is not restricted by the CFL, but is restricted to some extent by the nonlinear solver as the time step needs to be cut when the unpreconditioned GMRES is unable to solve the Jacobian system.…”