Ideal Magnetohydrodynamic Equations Research Articles

A Structure-Preserving Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at all Mach and Alfvén Numbers

We present a divergence-free semi-implicit finite volume scheme for the simulation of the ideal magnetohydrodynamics (MHD) equations which is stable for large time steps controlled by the local transport speed at all Mach and Alfvén numbers. An operator splitting technique allows to treat the convective terms explicitly while the hydrodynamic pressure and the magnetic field contributions are integrated implicitly, yielding two decoupled linear implicit systems. The linearity of the implicit part is achieved by means of a semi-implicit time linearization. This structure is favorable as second-order accuracy in time can be achieved relying on the class of semi-implicit IMplicit–EXplicit Runge–Kutta (IMEX-RK) methods. In space, implicit cell-centered finite difference operators are designed to discretely preserve the divergence-free property of the magnetic field on three-dimensional Cartesian meshes. The new scheme is also particularly well suited for low Mach number flows and for the incompressible limit of the MHD equations, since no explicit numerical dissipation is added to the implicit contribution and the time step is scale independent. Likewise, highly magnetized flows can benefit from the implicit treatment of the magnetic fluxes, hence improving the computational efficiency of the novel method. The convective terms undergo a shock-capturing second order finite volume discretization to guarantee the effectiveness of the proposed method even for high Mach number flows. The new scheme is benchmarked against a series of test cases for the ideal MHD equations addressing different acoustic and Alfvén Mach number regimes where the performance and the stability of the new scheme is assessed.

Cholla-MHD: An Exascale-capable Magnetohydrodynamic Extension to the Cholla Astrophysical Simulation Code

We present an extension of the massively parallel, GPU native, astrophysical hydrodynamics code Cholla to magnetohydrodynamics (MHD). Cholla solves the ideal MHD equations in their Eulerian form on a static Cartesian mesh utilizing the Van Leer + constrained transport integrator, the HLLD Riemann solver, and reconstruction methods at second and third order. Cholla’s MHD module can perform ≈260 million cell updates per GPU-second on an NVIDIA A100 while using the HLLD Riemann solver and second order reconstruction. The inherently parallel nature of GPUs combined with increased memory in new hardware allows Cholla’s MHD module to perform simulations with resolutions ∼5003 cells on a single high-end GPU (e.g., an NVIDIA A100 with 80 GB of memory). We employ GPU direct Message Passing Interface to attain excellent weak scaling on the exascale supercomputer Frontier, while using 74,088 GPUs and simulating a total grid size of over 7.2 trillion cells. A suite of test problems highlights the accuracy of Cholla’s MHD module and demonstrates that zero magnetic divergence in solutions is maintained to round off error. We also present new testing and CI tools using GoogleTest, GitHub Actions, and Jenkins that have made development more robust and accurate and ensure reliability in the future.

Hölder regularity of solutions and physical quantities for the ideal electron magnetohydrodynamic equations

In this paper, we make the first attempt to figure out the differences on Hölder regularity in time of solutions and conserved physical quantities between the ideal electron magnetohydrodynamic equations concerning Hall term and the incompressible Euler equations involving convection term. It is shown that the regularity in time of magnetic field B B is C t α 2 C_{t}^{\frac {\alpha }2} provided it belongs to L t ∞ C x α L_{t}^{\infty } C_{x}^{\alpha } for any α > 0 \alpha >0 , its energy is C t 2 α 2 − α C_{t}^{\frac {2\alpha }{2-\alpha }} as long as B B belongs to L t ∞ B ˙ 3 , ∞ α L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0 > α > 1 0>\alpha >1 and its magnetic helicity is C t 2 α + 1 2 − α C_{t}^{\frac {2\alpha +1}{2-\alpha }} supposing B B belongs to L t ∞ B ˙ 3 , ∞ α L_{t}^{\infty } \dot {B}^{\alpha }_{3,\infty } for any 0 > α > 1 2 0>\alpha >\frac 12 , which are quite different from the classical incompressible Euler equations.

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

A note on energy and cross‐helicity conservation in the ideal magnetohydrodynamic equations

In this paper, we are concerned with the conservation of total energy and cross‐helicity for the weak solutions in the ideal magnetohydrodynamic (MHD) equations. In the spirit of recent works of Berselli (J. Differ. Equ. 368 (2023), 350–375.) and Berselli‐Georgiadis (NoDEA Nonlinear Differ. Equ. Appl. 31 (2024), 33), by establishing a new generalized Constantin‐E‐Titi type commutator estimate to allow us to make full use of the total energy, we extend the previous classical results to a wider range of exponents. These results indicate the role of the time integrability, spatial integrability, and differential regularity of the velocity (magnetic field) in the conserved quantities of weak solutions in the ideal MHD equations.

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Three-dimensional Modeling of the Solar Wind and Local Interstellar Medium Interaction with Pickup Ions in the Presence of Heliospheric Current Sheet

Our three-dimensional, time-dependent, multi-fluid model has been used to investigate the solar wind (SW)–local interstellar medium (LISM) interaction with pickup ions (PUIs) treated as a separate fluid. A non-zero, but fixed, angle between the Sun’s magnetic and rotation axis is adopted. The flow of the plasma mixture (thermal SW protons, PUIs, and electrons), is described by the system of ideal magnetohydrodynamic equations with the source terms responsible for charge exchange between ions and neutral atoms. Different populations of neutral atoms are governed by the individual sets of the Euler equations. As the standard Rankine–Hugoniot relations are not appropriate to describe the anisotropic behavior of PUIs at the termination shock, we use a kinetically-derived set of boundary conditions at it. We extend our previous work [1] and perform these new simulations on a Cartesian grid. This approach allows us to maintain a uniform grid resolution in all directions, without compromising resolution, at large distances from the Sun. The possibility of transition of the SW flow to a stochastic regime in the region between the termination shock and heliopause is further investigated.

Numerical MHD models of stream interaction regions (SIRs) and corotating interaction regions (CIRs) using sunRunner3D: comparison with observations

ABSTRACT In this work, we present numerical simulations of Stream Interaction Regions (SIRs) and Corotating Interaction Regions (CIRs) using the sunrunner3d tool that employs as a coronal model the boundary conditions obtained by corhel/mas with the pluto code that describes the global 3D structure of the solar wind using the magnetohydrodynamics (MHD) approach in the inner heliosphere. Specifically, we selected a set of SIRs and CIRs observed by the Parker Solar Probe (PSP) and STEREO-A (STA) missions during the Carrington rotations (CRs) 2207 to 2210 and CRs from 2020 to 2022. In order to describe the dynamics of the plasma that constitutes the solar wind background conditions for the selected CRs, we solve the ideal MHD equations in an inertial frame of reference, managing the solar rotation by rotating the boundary values in ϕ (longitude) at a rate corresponding to the sidereal rotation rate of the solar equator. We show that our results using sunrunner3d can globally reproduce the plasma parameters, such as radial velocity, number proton density, and radial magnetic field strength of these large-scale structures, observed by PSP and STA at distances near the Sun and around 1 au, respectively. These results allow exploring the global evolution of SIRs/CIRs in the inner heliosphere using sunrunner3d.

Nambu Bracket for 3D Ideal Fluid Dynamics and Magnetohydrodynamics

Abstract The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by a Hamilton equation with respect to the Lie–Poisson bracket. The Nambu bracket manifestly represents the Lie–Poisson structure in terms of derivatives of the Casimir invariants. We construct a compact Nambu bracket representation for the 3D ideal MHD equations with the use of three Casimirs for the second Hamiltonians, the total entropy, and the magnetic and cross-helicities, whose coefficients are all constant. The Lie–Poisson bracket induced by this Nambu bracket does not coincide with the original one, but is supplemented by terms with an auxiliary variable. The supplemented Lie–Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether’s theorem, we show that the cross-helicity is an integral invariant associated with the particle-relabeling symmetry. Employing a Lagrange label function as the independent variable in the variational framework facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground as Noether’s theorem.

Numerical eigenanalysis of continuum geodesic acoustic mode by ideal magnetohydrodynamic model

In this study, we have developed a new eigenvalue code to solve the three-variables ideal magnetohydrodynamic (MHD) equation in flux coordinate system with the help of a symbolic vector analysis module and an automatic numerical discretization module, which were developed with the use of symbolic computation technique and were greatly facilitated the development of toroidal full MHD eigenvalue code SCELT. It has been used to solve for unstable modes and the present work is dedicated to the more difficult challenge of MHD continua, which ordinarily come with spatial singularities. With this new code, we were able to perform numerical calculations to determine the continuum spectrum of geodesic acoustic modes (GAMs), as well as obtain eigenfunctions for both the perturbed plasma displacement and perturbed magnetic field, incorporating a sufficient number of components. Through our global calculations, we have provided numerical evidence of singularities in the eigenfunctions, characterized by the type s−s0−1 and Ins−s0 . Moreover, we have examined the spatial variations of GAM fluctuations in regions both near and far from the singular surface. Here s represents a flux function that characterizes the magnetic flux surfaces and s=s0 defines the singular surface. Additionally, we have investigated the effects of finite aspect ratio, non-circular cross sections as well as the asymmetry on the frequency and eigenfunction of continuum GAMs through numerical analysis.

A Well-Balanced Semi-implicit IMEX Finite Volume Scheme for Ideal Magnetohydrodynamics at All Mach Numbers

We propose a second-order accurate semi-implicit and well-balanced finite volume scheme for the equations of ideal magnetohydrodynamics including gravitational source terms. The scheme treats all terms associated with the acoustic pressure implicitly while keeping the remaining terms part of the explicit sub-system. This semi-implicit approach makes the method particularly well suited for problems in the low Mach regime. We combine the semi-implicit scheme with the deviation well-balancing technique and prove that it maintains equilibrium solutions for the magnetohydrostatic case up to rounding errors. In order to preserve the divergence-free property of the magnetic field enforced by the solenoidal constraint, we incorporate a constrained transport method in the semi-implicit framework. Second order of accuracy is achieved by means of a standard spatial reconstruction technique with total variation diminishing property, and by an asymptotic preserving time stepping algorithm built upon the implicit-explicit Runge–Kutta time integrators. Numerical tests in the low Mach regime and near magnetohydrostatic equilibria support the low Mach and well-balanced properties of the numerical method.

Magnetic field suppression characteristics in interaction process between shock wave and light gas cylinder

Based on ideal compressible magnetohydrodynamics (MHD) equations, the interface instabilities induced by the interaction between planar shock wave and the light gas (Helium) cylinder under the influence of the magnetic fields with different directions are investigated numerically by using the CTU(corner transport upwind)+CT (constrained transport) algorithm. The numerical results elucidate the evolution of flow field characteristics and wave structures with and without magnetic field. Moreover, we examine the influence of the magnetic field direction on a characteristic scales (including the length, height and width of the central axis of gas cylinder), as well as the volume compressibility. Then, the mechanism of the magnetic field direction affecting the interface instability is studied in depth by integrating the analyses of the circulation, energy, velocity and magnetic force distribution within the flow field. The core of this study, is to explore the suppression mechanism of interface instability by magnetic field force. The results show that the magnetic pressure plays a crucial role in driving vorticity away from the interface, thereby reducing its deposition on the density interface. Simultaneously, it adheres to the divided vortex layer, thereby effectively isolating the influence of Richtmyer-Meshkov instability on the interface. On the other hand, the magnetic tension adheres to the separated vortex layer, and its direction is opposite to that of the vorticity generated by the shear of interface velocity. This action effectively suppresses the Kelvin-Helmholtz instability and the rolling-up of vortices on the density interface. Additionally, under the control of a longitudinal magnetic field, the direction of magnetic tension is opposite to the direction of the central jet, effectively suppressing the development of Rayleigh-Taylor instability.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

A novel flux splitting based on wave-particle splitting for ideal magnetohydrodynamics

A novel flux splitting method based on wave-particle splitting is developed for one-dimensional ideal magnetohydrodynamics. While ideal magnetohydrodynamics (MHD) equations are non-convex with non-homogeneous flux as opposed to their hydrodynamic counterparts, the present flux splitting methods cannot develop Riemann solver. The proposed approach based on wave-particle method referred as Advection Magnetic-direction Wave Particle Splitting (AMWPS) scheme splits the flux into four parts: the wave-like total pressure part, the wave-like normal magnetic part, the particle-like advection part and the particle-like tangential magnetic part. The particle-like parts can be solved by the scheme developed by Toro and Vazquez-Cendon (TV scheme) and we propose a novel approach to the solution of Riemann problem formed by the wave-like parts including total pressure sub-flux and normal magnetic sub-flux. As for the ordered wave foliation in MHD and other limitations in wave structure, we also analyze the relation between plasma β and specific heat ratio γ. The advantage of AMWPS is that it can sharply capture isolated, stationary Alfven wave discontinuity. Several one-dimensional and two-dimensional MHD problems have been tested to highlight the accuracy, positivity preservation and robustness of AMWPS scheme and comparative studies show that AMWPS significantly outperforms the Riemann solver named Harten, Lax and Leer for contact wave (HLLC) in most cases.

Local well-posedness of the free-boundary incompressible magnetohydrodynamics with surface tension

We prove the local well-posedness of the 3D free-boundary incompressible ideal magnetohydrodynamics (MHD) equations with surface tension, which describe the motion of a perfect conducting fluid in an electromagnetic field. We adapt the ideas developed in the remarkable paper [11] by Coutand and Shkoller to generate an approximate problem with artificial viscosity indexed by κ>0 whose solution converges to that of the MHD equations as κ→0. However, the local well-posedness of the MHD equations is no easy consequence of Euler equations thanks to the strong coupling between the velocity and magnetic fields. This paper is the continuation of the second and third authors' previous work [38] in which the a priori energy estimate for incompressible free-boundary MHD with surface tension is established. But the existence is not a trivial consequence of the a priori estimate as it cannot be adapted directly to the approximate problem due to the loss of the symmetric structure.

Incompressible Limit of Ideal Magnetohydrodynamic Equations with a Perfectly Conducting Wall Condition

Incompressible Limit of Ideal Magnetohydrodynamic Equations with a Perfectly Conducting Wall Condition

Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.

Small Alfvén number limit for shallow water magnetohydrodynamics

Small Alfvén number limit of the shallow water magnetohydrodynamic equations with ill-prepared initial data is proved by applying fast averaging method. It is rigorously justified that the solutions of the original system tend to the solution of a one-dimensional system coupled with a linear system as the Alfvén number goes to zero. The approach is also used to study the small Alfvén number limit of the three-dimensional ideal incompressible magnetohydrodynamic equations.

Incompressible limit of nonisentropic ideal magnetohydrodynamic equations with periodic boundary conditions

This paper is concerned with the incompressible limit of the non-isentropic ideal magnetohydrodynamic equations with periodic boundary conditions and general initial data. Weak convergence results of the smooth solutions are established by filtering method in the torus.

Numerical path preserving Godunov schemes for hyperbolic systems

This paper primarily concerns the discontinuities capturing problems in nonconservative and nonconvex conservative hyperbolic systems. For the Godunov scheme of nonconservative hyperbolic systems, the numerical dissipation at discontinuous points in the simulation process is analyzed grid point by grid point through a new perspective. The numerical paths implied in the nonconservative variables represent different averaging and dissipating processes from the conservative cases. Unphysical dissipation of nonconservative variables ruins Rankine-Hugoniot relations, contributing to incorrect jumps, wrong propagation speed, and spurious fluctuations in other characteristic fields. For the discontinuities capturing problem of nonconservative hyperbolic systems, a novel numerical path preserving (NPP) method is proposed to modify the original Godunov schemes so that the dissipation of the numerical methods at discontinuities is carried out strictly following the consistent numerical path. Numerical simulations of the nonconservative systems are performed for isothermal and Euler equations. The results indicate that the NPP method can correctly capture the discontinuous structures and verify the correctness of our theoretical analysis. Additionally, the NPP method is extended to nonconvex hyperbolic conservation systems, and the Alfvénic wave (discontinuity) of the one-dimensional ideal magnetohydrodynamic (MHD) equations is simulated. It is found that the nonconvex nature of the flux causes unphysical compound wave structures while simulating the Alfvénic wave with the popular schemes nowadays, e.g., Roe of flux difference splitting method and WENO with flux vector splitting method. Modifying the original Godunov schemes with the NPP method proposed in this paper ensures correct simulation of Alfvénic discontinuity in MHD equations. It validates the effectiveness of the NPP method for nonconvex hyperbolic conservation systems.