The equations of magnetohydrodynamics are generally known to be difficult to solve numerically. They are highly nonlinear and exhibit strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers.

In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the single-fluid incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances.

We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.

The governing equations for magnetohydrodynamics (MHD) possess a number of conserved quantities that are difficult to preserve in numerical discretizations. In recent work, we constructed a finite element method for inhomogeneous, incompressible MHD that preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, incompressibility, and div(B) = 0 to machine precision. This talk will summarize the method and discuss extensions to compressible and resistive MHD. Special attention will be paid to the derivation of the method via variational principles and to the treatment of variable density.

The Euler equations governing the flow of an incompressible ideal fluid can be recast as a transport problem for a weakly closed time-dependent 1-form, where the velocity field is associated with the 1-form through the Euclidean Hodge operator. This formulation is amenable to semi-Lagrangian discretization in time combined with a finite-element Galerkin discretization in space relying on discrete differential forms as provided by finite-element exterior calculus (FEEC). We elaborate the details of a first and second-order realization of an interpolatory semi-Lagrangian scheme based on so-called small edges in the second-order case. We propose a semi-implicit treatment of the solution-dependent flow and add constraints enforcing the conservation of total energy and helicity.

Our focus is on derivations of the schemes, details of implementation, and thorough numerical tests. The latter confirm the excellent stability properties of the schemes and the expected asymptotic convergence. We also observe robustness with respect to the strength of an added diffusion term.

We propose a new electrostatic particle-in-cell algorithm able to use large timesteps compared to particle gyro-period under a (to begin, uniform) large external magnetic field. The algorithm extends earlier electrostatic fully implicit PIC implementations [1] with a new asymptotic-preserving (AP) particle-push scheme [2] that allows timesteps much larger than particle gyroperiods. In the large-timestep limit, the AP integrator preserves all the averaged particle drifts, while recovering particle full orbits with small timesteps. The scheme allows for a seamless, efficient treatment of coexisting magnetized and unmagnetized particles, conserves energy and charge exactly, and does not spoil implicit solver performance. Key to the approach is the generalization of the particle substepping approach introduced in Ref. [1] to allow for orbit segments of each substep much larger than cell sizes. The uniform-magnetic-field condition allows us to use the standard Crank-Nicolson update [1] without modification [2], which is a necessary preliminary step to demonstrate the viability of the approach for more general magnetic field topologies (which will otherwise require the general algorithm proposed in Ref. [2]). We demonstrate by numerical experiment with several strongly magnetized problems (diocotron instability, modified two-stream instability, and drift-wave instability) that two orders of magnitude wall-clock-time speedups are possible vs. the standard fully implicit electrostatic PIC algorithm without sacrificing solution quality and while preserving strict charge and energy conservation.

[1] Chen, Guangye, Luis Chacón, and Daniel C. Barnes. "An energy-and charge-conserving, implicit, electrostatic particle-in-cell algorithm." Journal of Computational Physics 230.18 (2011): 7018-7036.

[2] Ricketson, Lee F., and Luis Chacón. "An energy-conserving and asymptotic-preserving charged-particle orbit implicit time integrator for arbitrary electromagnetic fields." Journal of Computational Physics 418 (2020): 109639.