Block preconditioners for energy stable schemes of magnetohydrodynamics equations

In this article, we develop the first and second order unconditionally energy stable schemes for magnetohydrodynamics (MHD) equations and design robust preconditioners for these schemes. Inspired by operator preconditioning ideas, appropriate parameter-dependent norms on function spaces are subtly defined to uniformly bound the bilinear and trilinear terms, which implies the uniform well-posedness of the schemes under the newly defined norms. Then robust block preconditioners are constructed using the Riesz operators. We prove that, if time step size k <= C$$ k\le C $$, the proposed preconditioners are uniformly robust with respect to physical parameters and discrete parameters. Various numerical experiments, including energy stability tests, Kelvin-Helmholtz instability and magnetic driven cavity physical benchmark problems, are presented to manifest unconditional energy stability of the schemes and robustness of our preconditioners.