The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form $\Delta t\propto (\Delta x)^2$ for explicit calculations. A new semi--implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It it based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.
Deep Dive into A semi-implicit Hall-MHD solver using whistler wave preconditioning.
The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form $\Delta t\propto (\Delta x)^2$ for explicit calculations. A new semi–implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It it based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method.
In many space-, astrophysical and high temperature plasma systems collisions do not play the most important role in describing the departure from the ideal magnetohydrodynamics (MHD)
where ρ, v, B and p denote mass density, velocity, magnetic field and pressure, respectively. Typical examples include filamentation and singularity formation, collisionless reconnection and collisionless shocks [1,2,3,4,5,6,7,8,9].
Therefore, on scales smaller than the ion inertia length additional processes have to be taken into account in a generalized Ohm’s law
Numerically, the most difficult term is the Hall-term:
It allows for whistler wave solutions with a quadratic dispersion relation and thus poses a severe time step restriction for a temporal explicit discretisation.
To introduce our treatment of the Hall-term, we simplify our system and use only this electric field in the induction equation which decouples it from the other part of the MHD equations and yields the following nonlinear equation
Solutions of the linearized equations are the whistler waves mentioned above which satisfy the dispersion relation ω
, for a constant density ρ and a guiding field magnitude | B|. Numerical approaches using explicit schemes applied to this equation must ensure that the chosen time step fulfills ∆t ∝ (∆x) 2 , due to the Courant-Levy-Friedrichs criterion -∆x denoting the grid spacing. The CFL number is given by the ratio of the phase velocity to the grid velocity
where k = k max = 2π ∆x is the maximum wave number. Thus resolving small structures, e.g. the reconnection zone, results in large computation times, due to the unavoidable small time steps.
Implicit schemes allow to avoid this restrictive condition by providing unconditional numerical stability. Much progress on implicit solvers has been done by Harned and Mikić [10] and Chacón and Knoll [11]. However, the approach of [10] requires a guiding magnetic field and the approach of [11] can’t easily be adopted for simulations with adaptive mesh refinements [12,13,14,15], although work in this direction is in progress.
Here we present a simple physics based semi-implicit Crank-Nicolson type scheme which due to its locality properties is suitable for parallel computations as well as for use in adaptive mesh refinement simulations. This physics based solver uses a whistler wave decomposition to accelerate the fix-point iteration. Due to its convergence properties it can act as a smoother for a nonlinear multigrid scheme.
The first part of this paper presents the general numerical method which then is specialized to one dimension. This allows us to show analytically its convergence. After that the nonlinear two-dimensional case and its convergence are presented, while in the last section our method is used to solve a twodimensional reconnection problem.
The Richardson iteration [16] is the base of our solver. A Richardson iteration is the most general fix point iteration for a nonlinear equation F ( x) = 0
where k is the iteration index. Given a contractive map K, the x k converge in the limit k → ∞. The rate of convergence will in general depend on α. The main task is to find a suitable preconditioner adapted to the Hall-term. This can be realized as a matrix P
In the special case of the Newton iteration P is the inverse of the Jacobi matrix of F . Here, we try to find a physics based preconditioner which is more local and thus suitable for parallel and block-adaptive calculations.
For the Crank-Nicolson type discretisation, we obtain
with
and where B n is the magnetic field taken at the time step n (time steps are indicated by the first upper index). The equation to be solved reads now
Its solution with a given B n is the magnetic field at the next time step n + 1.
To determine a solution we iterate equation (11) following the method given by (8). At this point we introduce an additional upper index which defines the iteration step. So that B n+1,k is k-th iteration of the magnetic field for the time step n + 1.
As mentioned above, the important point in this iteration is the preconditioning. To motivate our preconditioner, we start with the one-dimensional version of eqn. (7), where B depends only on x. In the one-dimensional case, we have the special situation that the fix-point problem reduces to a linear one. In this case, our physics based preconditioner reduces to the standard ω-Jacobi iteration. In more than one dimensions the situation is genuinely nonlinear but the smoothing properties of our preconditioner are still similar to that of a Jacobi iteration for linear problems.
3 Linear 1D Case
Considering only the x direction results in the following system of equations
The x component of B is initially set to a constant B 0 in space and stays constant. The discretised equations for F are needed for the iteration. Following equation (11) and again using B * = 1 2 B n + B n+1 these are given for each grid point i
To calculate the next time step B n+1
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