Ideal Compressible Magnetohydrodynamics Research Articles

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.

The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with Perfectly Conducting Boundary

The Incompressible Limit of the Equations of Compressible Ideal Magneto-Hydrodynamics with Perfectly Conducting Boundary

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.

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.

On local well-posedness of problems with characteristic free boundary for hyperbolic systems of conservation laws

The proof of the local-in-time existence and uniqueness of a smooth solution to a free boundary problem for a hyperbolic system of conservation laws has some additional difficulties if the free boundary is a characteristic of this system. They are connected with the loss of control of the normal derivatives and the possible non-ellipticity of the symbol of the free boundary. Another peculiarity of problems with characteristic free boundary is that usually a loss of derivatives of the coefficients and source terms occurs in a priori estimates for the corresponding linearized problems. Moreover, the boundary conditions in the linearized problem can be non-dissipative, which makes it difficult to use the energy method. We describe methods for overcoming these difficulties. Our main examples are free boundary problems for Euler's equations and the equations of ideal compressible magnetohydrodynamics, for which we review the results on their local well-posedness. Bibliography: 61 titles.

Existence of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics

AbstractWe establish the local existence and uniqueness of multi‐dimensional contact discontinuities for the ideal compressible magnetohydrodynamics (MHD) in Sobolev spaces, which are most typical interfacial waves for astrophysical plasmas and prototypical fundamental waves for hyperbolic systems of conservation laws. Such waves are characteristic discontinuities for which there is no flow across the discontinuity surface while the magnetic field crosses transversely, which lead to a two‐phase free boundary problem where the pressure, velocity and magnetic field are continuous across the interface whereas the entropy and density may have jumps. To overcome the difficulties of possible nonlinear Rayleigh–Taylor instability and loss of derivatives, here we use crucially the Lagrangian formulation and Cauchy's celebrated integral (1815) for the magnetic field. These motivate us to define two special good unknowns; one enables us to capture the boundary regularizing effect of the transversal magnetic field on the flow map, and the other one allows us to get around the troublesome boundary integrals due to the transversality of the magnetic field. In particular, our result removes the additional assumption of the Rayleigh–Taylor sign condition required by Morando, Trakhinin and Trebeschi (J. Differ. Equ. 258 (2015), no. 7, 2531–2571; Arch. Ration. Mech. Anal. 228 (2018), no. 2, 697–742) and holds for both 2D and 3D and hence gives a complete answer to the two open questions raised therein. Moreover, there is no loss of derivatives in our well‐posedness theory. The solution is constructed as the inviscid limit of solutions to some suitably‐chosen nonlinear approximate problems for the two‐phase compressible viscous non‐resistive MHD.

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

On vacuum free boundary problems in ideal compressible magnetohydrodynamics

AbstractWe survey some recent results on vacuum free boundary problems in three‐dimensional ideal compressible magnetohydrodynamics, restate the main theorems in our works (Secchi and Trakhinin, Nonlinearity 27(1) (2014) 105–169; Trakhinin and Wang, Arch. Ration. Mech. Anal. 239(2) (2021) 1131–1176; Trakhinin and Wang, Math. Ann. 383(1–2) (2022) 761–808; Trakhinin and Wang, SIAM J. Math. Anal. 54(6) (2022) 5888–5921), and provide an alternative proof for the linear well‐posedness of the full plasma–vacuum interface problem under the noncollinearity condition.

Nonlinear Stability and Existence of Two-Dimensional Compressible Current-Vortex Sheets

We are concerned with the nonlinear stability and existence of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. This is a nonlinear hyperbolic initial-boundary value problem with a characteristic free boundary. It is well-known that current-vortex sheets may be at most weakly (neutrally) stable due to the existence of surface waves solutions that yield a loss of derivatives in the energy estimate of the solution with respect to the source terms. We first identify a sufficient condition ensuring the weak stability of the linearized current-vortex sheets problem. Under this stability condition for the background state, we show that the linearized problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Based on the weakly linear stability results, we then establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash–Moser iteration, provided that the stability condition is satisfied at each point of the initial discontinuity. This result gives a new confirmation of the stabilizing effect of sufficiently strong magnetic fields on Kelvin–Helmholtz instabilities.

Suppression mechanism of Richtmyer–Meshkov instability by transverse magnetic field with different strengths

The Richtmyer–Meshkov instability (RMI) is caused by an incident planar shock wave impinging on the heavy-gas-density interface. We have numerically investigated the RMI controlled by different transverse magnetic-field strengths based on the ideal compressible magnetohydrodynamics (MHD) equations. The MHD equations are solved by the corner transport upwind + constrained transport algorithm, which guarantees a divergence-free constraint on the magnetic field. We discuss the flow characteristics and shock patterns in both classical hydrodynamic and MHD situations and verify our conclusions by comparing the experimental results with the numerical results. The results show that the magnetic field modifies the pressure-gradient distribution, and the baroclinic vorticity splits and attaches to the MHD shock waves. In addition, the results indicate that the interaction of shock wave and density interface changes the distribution of magnetic-field energy and distorts the magnetic induction line in the region of magnetic-field energy accumulation. The distortion of the magnetic induction lines alters the magnetic field gradient and creates a magnetic tension that produces a torque opposing that generated by the shear force on the vorticity layer, so the Kelvin–Helmholtz instability is effectively suppressed and no Kelvin–Helmholtz vortex appears on the vorticity layer. The result is that the interface instability is suppressed.

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash--Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

Regularity and energy conservation for the compressible MHD equations

In this paper, we aRemark concerned with the regularity and energy conservation of the weak solutions for inhomogeneous compressible ideal MHD (magnetohydrodynamics) systems. We give two sufficient 条件 on the regularity of solutions in order to ensuRemark the energy to be conserved, wheRemark $(t,~x)\in~(0,~T)\times\mathbb{T}^d$ $(d\geq~1)$. The main tool we have employed relies on the commutator estimates similar to those used by Constantin et al. (1994) for the homogeneous incompressible Euler equations.

High order well-balanced finite volume methods for multi-dimensional systems of hyperbolic balance laws

We introduce a general framework for the construction of well-balanced finite volume methods for hyperbolic balance laws. We use the phrase well-balancing in a broader sense, since our proposed method can be applied to exactly follow any solution of any system of hyperbolic balance laws in multiple spatial dimensions and not only time independent solutions. The solution has to be known a priori, either as an analytical expression or as discrete data. The proposed framework modifies the standard finite volume approach such that the well-balancing property is obtained and in case the method is high order accurate, this is maintained under our modification. We present numerical tests for the compressible Euler equations with and without gravity source term and with different equations of state, and for the equations of compressible ideal magnetohydrodynamics.

Convergence rate estimates for the low Mach and Alfvén number three-scale singular limit of compressible ideal magnetohydrodynamics

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfvén numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors’ previous paper, is much more complicated than for the classical two-scale singular limits.

Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma--vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh--Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash--Moser iteration scheme. Our result is uniform in the light speed and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.

Inverse transfer of magnetic helicity in supersonic magnetohydrodynamic turbulence

The inverse transfer of magnetic helicity is studied through a fourth-order finite volume numerical scheme in the framework of compressible ideal magnetohydrodynamics (MHD), with an isothermal equation of state. Using either a purely solenoidal or purely compressive mechanical driving, a hydrodynamic turbulent steady-state is reached, to which small-scale magnetic helical fluctuations are injected. The steady-state root mean squared Mach numbers considered range from 0.1 to about 11. In all cases, a growth of magnetic structures is observed. While the measured magnetic helicity spectral scaling exponents are similar to the one measured in the incompressible case for the solenoidally-driven runs, significant deviations are observed even at relatively low Mach numbers when using a compressive driving. A tendency towards equipartition between the magnetic and kinetic fields in terms of energy and helicity is noted. The joint use of the helical decomposition in the framework of shell-to-shell transfer analysis reveals the presence of three distinct features in the global picture of a magnetic helicity inverse transfer. Those are individually associated with specific scale ranges of the advecting velocity field and commensurate helical contributions.

A positivity-preserving Lagrangian Discontinuous Galerkin method for ideal magnetohydrodynamics equations in one-dimension

In this paper, we propose a conservative Lagrangian Discontinuous Galerkin (DG) scheme for solving the ideal compressible magnetohydrodynamics (MHD) equations in one-dimensional. This scheme can preserve positivity of physically positive variables such as density and thermal pressure. We first develop a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver a first order positivity-preserving Lagrangian DG scheme can be constructed. Then we design a high order positivity-preserving and conservative Lagrangian DG scheme by using the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter. We adopt TVB minmod limiter with the local characteristic fields for the ideal MHD system to control spurious oscillations around the shock wave. Some numerical examples are presented to demonstrate the accuracy and positivity-preserving property of our scheme.

Implicit hybridized discontinuous Galerkin methods for compressible magnetohydrodynamics

We present hybridized discontinuous Galerkin (HDG) methods for ideal and resistive compressible magnetohydrodynamics (MHD). The HDG methods are fully implicit, high-order accurate and endowed with a unique feature which distinguishes themselves from other discontinuous Galerkin (DG) methods. In particular, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby resulting in considerably smaller global degrees of freedom than other DG methods. Furthermore, we develop a shock capturing method to deal with shocks by appropriately adding artificial bulk viscosity, molecular viscosity, thermal conductivity, and electric resistivity to the physical viscosities in the MHD equations. We show the optimal convergence of the HDG methods for ideal MHD problems and validate our resistive implementation for a magnetic reconnection problem. For smooth problems, we observe that employing a generalized Lagrange multiplier (GLM) formulation can reduce the errors in the divergence of the magnetic field by two orders of magnitude. We demonstrate the robustness of our shock capturing method on a number of test cases and compare our results, both qualitatively and quantitatively, with other MHD solvers. For shock problems, we observe that an effective treatment of both the shock wave and the divergence-free constraint is crucial to ensuring numerical stability.

Extension of the subgrid-scale gradient model for compressible magnetohydrodynamics turbulent instabilities

Performing accurate large eddy simulations in compressible, turbulent magnetohydrodynamics (MHDs) is more challenging than in nonmagnetized fluids due to the complex interplay between kinetic, magnetic, and internal energy at different scales. Here, we extend the subgrid-scale gradient model, so far used in the momentum and induction equations, to also account for the unresolved scales in the energy evolution equation of a compressible ideal MHD fluid with a generic equation of state. We assess the model by considering box simulations of the turbulence triggered across a shear layer by the Kelvin-Helmholtz instability, testing cases where the small-scale dynamics cannot be fully captured by the resolution considered, such that the efficiency of the simulated dynamo effect depends on the resolution employed. This lack of numerical convergence is actually a currently common issue in several astrophysical problems, where the integral and fastest-growing-instability scales are too far apart to be fully covered numerically. We perform a priori and a posteriori tests of the extended gradient model. In the former, we find that, for many different initial conditions and resolutions, the gradient model outperforms other commonly used models in terms of correlation with the residuals coming from the filtering of a high-resolution run. In the second test, we show how a low-resolution run with the gradient model is able to quantitatively reproduce the evolution of the magnetic energy (the integrated value and the spectral distribution) coming from higher-resolution runs. This extension is the first step toward the implementation in relativistic MHDs.

Singular limits of the equations of compressible ideal magneto-hydrodynamics in a domain with boundaries

Singular limits of the equations of compressible ideal magneto-hydrodynamics in a domain with boundaries