Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton–Krylov–Schwarz

Numerical simulations of the four-field extended magnetohydrodynamics (MHD) equations with hyper-resistivity terms present a difficult challenge because of demanding spatial resolution requirements. A time-dependent sequence of r-refinement adaptive grids obtained from solving a single Monge–Ampère (MA) equation addresses the high-resolution requirements near the x-point for numerical simulation of the magnetic reconnection problem. The MHD equations are transformed from Cartesian coordinates to solution-defined curvilinear coordinates. After the application of an implicit scheme to the time-dependent problem, the parallel Newton–Krylov–Schwarz (NKS) algorithm is used to solve the system at each time step. Convergence and accuracy studies show that the curvilinear solution requires less computational effort than a pure Cartesian treatment. This is due both to the more optimal placement of the grid points and to the improved convergence of the implicit solver, nonlinearly and linearly. The latter effect, which is significant (more than an order of magnitude in number of inner linear iterations for equivalent accuracy), does not yet seem to be widely appreciated.