Hyperbolic Divergence Cleaning Research Articles

An Extension of Gmunu: General-relativistic Resistive Magnetohydrodynamics Based on Staggered-meshed Constrained Transport with Elliptic Cleaning

We present the implementation of general-relativistic resistive magnetohydrodynamics solvers and three divergence-free handling approaches adopted in the General-relativistic multigrid numerical (Gmunu) code. In particular, implicit–explicit Runge–Kutta schemes are used to deal with the stiff terms in the evolution equations for small resistivity. The three divergence-free handling methods are (i) hyperbolic divergence cleaning (also known as the generalized Lagrange multiplier), (ii) staggered-meshed constrained transport schemes, and (iii) elliptic cleaning through a multigrid solver, which is applicable in both cell-centered and face-centered (stagger grid) magnetic fields. The implementation has been tested with a number of numerical benchmarks from special-relativistic to general-relativistic cases. We demonstrate that our code can robustly recover from the ideal magnetohydrodynamics limit to a highly resistive limit. We also illustrate the applications in modeling magnetized neutron stars, and compare how different divergence-free handling methods affect the evolution of the stars. Furthermore, we show that the preservation of the divergence-free condition of the magnetic field when using staggered-meshed constrained transport schemes can be significantly improved by applying elliptic cleaning.

Fully-implicit finite volume method for the ideal two-fluid plasma model

We present a novel numerical model that simulates ideal two-fluid plasmas coupled to the full set of Maxwell’s equations with application to space and laboratory plasmas. We use a fully-implicit finite volume method for unstructured meshes, that uses an advection upstream splitting method (i.e., AUSM+-up) for all speeds to discretize the numerical fluxes of the fluids. In addition, we discretize Maxwell’s equations with a modified-Rusanov scheme. The electromagnetic numerical dissipation is scaled using the scales of the fluid-electromagnetics coupled problem that are found to be very different from those of the uncoupled problem. Our numerical scheme guarantees that the elliptical constraints of Maxwell’s equations are satisfied by using hyperbolic divergence cleaning. We validate the performance and accuracy of our model by simulating the following conventional cases: a circularly polarized wave, a Brio–Wu type shock tube, and a two-fluid plasma reconnection with the GEM challenge set up. Our model reveals the complexity of the two-fluid model compared to magnetohydrodynamics (MHD) models, as the inclusion of charge separation, the displacement current and the electron dynamics present are ignored by the MHD simplifications. The two-fluid model shows the presence of electromagnetic and plasma waves and the effect that they have in even the simplest cases. We also compare our model to other available two-fluid models and find our results to be in good agreement.

Complex-Frequency Shifted PMLs for Maxwell’s Equations With Hyperbolic Divergence Cleaning and Their Application in Particle-in-Cell Codes

The simulation of unbounded domains inevitably requires an artificial truncation of the computational domain and spurious reflections resulting from this procedure are a common problem. In this paper, a perfectly matched layer formulation for Maxwell’s equations in purely hyperbolic form is presented. The model is applied to standard wave attenuation problems and particle-in-cell simulations of electron beam devices.

Roe-type schemes for shallow water magnetohydrodynamics with hyperbolic divergence cleaning

We discuss Roe-type linearizations for non-conservative shallow water magnetohydrodynamics without and with hyperbolic divergence cleaning.

Hyperbolic divergence cleaning, the electrostatic limit, and potential boundary conditions for particle-in-cell codes

In a numerical solution of the Maxwell–Vlasov system, the consistency with the charge conservation and divergence conditions has to be kept solving the hyperbolic evolution equations of the Maxwell system, since the vector identity ∇⋅(∇×u→)=0 and/or the charge conservation of moving particles may be not satisfied completely due to discretization errors. One possible method to force the consistency is the hyperbolic divergence cleaning. This hyperbolic constraint formulation of Maxwell's equations has been proposed previously, coupling the divergence conditions to the hyperbolic evolution equations, which can then be treated with the same numerical method. We pick up this method again and show that electrostatic limit may be obtained by accentuating the divergence cleaning sub-system and converging to steady state. Hence, the electrostatic case can be treated by the electrodynamic code with reduced computational effort. In addition, potential boundary conditions as often given in practical applications can be coupled in a similar way to get appropriate boundary conditions for the field equations. Numerical results are shown for an electric dipole, a parallel-plate capacitor, and a Langmuir wave. The use of potential boundary conditions is demonstrated in an Einzel lens simulation.

Relativistic MHD with adaptive mesh refinement

This paper presents a new computer code to solve the general relativistic magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh refinement (AMR). The fluid equations are solved using a finite difference convex ENO method (CENO) in 3 + 1 dimensions, and the AMR is Berger–Oliger. Hyperbolic divergence cleaning is used to control the ∇ ⋅ B = 0 constraint. We present results from three flat space tests, and examine the accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel solution. The AMR simulations substantially improve performance while reproducing the resolution equivalent unigrid simulation results. Finally, we discuss strong scaling results for parallel unigrid and AMR runs.

Relativistic MHD and excision: formulation and initial tests

A new algorithm for solving the general relativistic MHD equations is described in this paper. We design our scheme to incorporate black hole excision with smooth boundaries, and to simplify solving the combined Einstein and MHD equations with AMR. The fluid equations are solved using a finite difference convex ENO method. Excision is implemented using overlapping grids. Elliptic and hyperbolic divergence cleaning techniques allow for maximum flexibility in choosing coordinate systems, and we compare both methods for a standard problem. Numerical results of standard test problems are presented in two-dimensional flat space using excision, overlapping grids and elliptic and hyperbolic divergence cleaning.

Hyperbolic Divergence Cleaning for the MHD Equations

In simulations of magnetohydrodynamic (MHD) processes the violation of the divergence constraint causes severe stability problems. In this paper we develop and test a new approach to the stabilization of numerical schemes. Our technique can be easily implemented in any existing code since there is no need to modify the solver for the MHD equations. It is based on a modified system in which the divergence constraint is coupled with the conservation laws by introducing a generalized Lagrange multiplier. We suggest a formulation in which the divergence errors are transported to the domain boundaries with the maximal admissible speed and are damped at the same time. This corrected system is hyperbolic and the density, momentum, magnetic induction, and total energy density are still conserved. In comparison to results obtained without correction or with the standard “divergence source terms,” our approach seems to yield more robust schemes with significantly smaller divergence errors.