A novel structure preserving semi-implicit finite volume method for viscous and resistive magnetohydrodynamics. (arXiv:2012.11218v2 [math.NA] UPDATED)
<a href="http://arxiv.org/find/math/1/au:+Fambri_F/0/1/0/all/0/1">Francesco Fambri</a>
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.
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.
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