MHD Equations Research Articles

Fully decoupled, linearized and stabilized finite volume method for the time-dependent incompressible MHD equations

In this paper, we consider the stability and convergence of the fully decoupled and linearized numerical scheme for the time-dependent incompressible magnetohydrodynamic equations based on the finite volume method. The lowest equal-order mixed finite element pair (P1-P1-P1) is used to approximate the velocity, pressure and magnetic fields, and the pressure projection stabilization is introduced to bypass the restriction of the discrete inf-sup condition. The semi-implicit treatment is used to linearize the nonlinear terms and the first order projection scheme is adopted to split the velocity and pressure, then a series of fully decoupled and linearized subproblems are formed. From the view of theoretical analysis, some novel stability results of the numerical solutions in both spatial semi-discrete scheme and time–space fully discrete scheme are provided, the optimal error estimates are also presented by using the energy method and choosing different test functions. From the view of computational results, the fully decoupled and linearized numerical scheme not only keeps good accuracy, but also saves a lot of computational cost.

Global well-posedness for 2D non-resistive MHD equations in half-space

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the non-resistive MHD equations have a unique global strong solution around the equilibrium state (0, e1) for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

Global well-posedness for the inhomogeneous incompressible MHD equations with variable viscosity coefficient and electrical conductivity

In this paper, we study the Cauchy problem for the inhomogeneous incompressible MHD system with variable viscosity coefficient and electrical conductivity. It is shown that the 3-D inhomogeneous incompressible MHD system has unique global solutions with small initial data conditions. Furthermore, we establish the global well-posedness of the two-dimensional MHD equations in critical spaces, given certain conditions on the density.

Space Weather with Quantified Uncertainties: Improving Space Weather Predictions with Data-Driven Models of the Solar Atmosphere and Inner Heliosphere

To address Objective II of the National Space Weather Strategy and Action Plan “Develop and Disseminate Accurate and Timely Space Weather Characterization and Forecasts” and US Congress PROSWIFT Act 116–181, our team is developing a new set of open-source software that would ensure substantial improvements of Space Weather (SWx) predictions. On the one hand, our focus is on the development of data-driven solar wind models. On the other hand, each individual component of our software is designed to have accuracy higher than any existing SWx prediction tools with a dramatically improved performance. This is done by the application of new computational technologies and enhanced data sources. The development of such software paves way for improved SWx predictions accompanied with an appropriate uncertainty quantification. This makes it possible to forecast hazardous SWx effects on the space-borne and ground-based technological systems, and on human health. Our models include (1) a new, open-source solar magnetic flux model (OFT), which evolves information to the back side of the Sun and its poles, and updates the model flux with new observations using data assimilation methods; (2) a new potential field solver (POT3D) associated with the Wang–Sheeley–Arge coronal model, and (3) a new adaptive, 4-th order of accuracy solver (HelioCubed) for the Reynolds-averaged MHD equations implemented on mapped multiblock grids (cubed spheres). We describe the software and results obtained with it, including the application of machine learning to modeling coronal mass ejections, which makes it possible to improve SWx predictions by decreasing the time-of-arrival mismatch. The tests show that our software is formally more accurate and performs much faster than its predecessors used for SWx predictions.

High-order genuinely multidimensional finite volume methods via kernel-based WENO

In this paper a family of fully multidimensional kernel-based reconstruction schemes for use in finite volume methods (FVMs) will be presented. These methods are intended for use in shock dominated problems, and stability is achieved through a suitable adaptation of the Adaptive Order Weighted Essentially Non-Oscillatory (WENO-AO) method to the proposed kernel-based reconstruction schemes. There are a number of key difficulties in the design of high-order finite volume schemes which will be discussed and addressed. High (4th and 6th) order convergence will be demonstrated on smooth exact solutions of the ideal MHD equations. The very same scheme will then be applied to extremely stringent astrophysical benchmark problems.

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

Global well-posedness and optimal decay for incompressible MHD equations with fractional dissipation and magnetic diffusion

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

High Order Structure-Preserving Finite Difference WENO Schemes for MHD Equations with Gravitation in all Sonic Mach Numbers

Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations

In this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be successfully applied to guarantee the existence of (weak) solutions. To achieve this, the quaternionic integral operator calculus serves as a key ingredient allowing us to apply Schauder’s fixed point theorem. The advantage of the approach using Schauder’s fixed point theorem is that it is also applicable to large data since it does not require any kind of contraction property. These considerations will allow us to provide explicit iterative algorithms for its numerical solution. Finally to obtain more precise a-priori estimates one can use in the situations dealing with small data the Banach fixed point theorem which then also grants the uniqueness.

A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations

A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations

The global existence and analyticity of a mild solution to the 3D regularized MHD equations

The global existence and analyticity of a mild solution to the 3D regularized MHD equations

On the Sobolev stability threshold for shear flows near Couette in 2D MHD equations

In this work, we study the Sobolev stability of shear flows near Couette in the 2D incompressible magnetohydrodynamics (MHD) equations with background magnetic field $(\alpha,0 )^\top$ on $\mathbb {T}\times \mathbb {R}$ . More precisely, for sufficiently large $\alpha$ , we show that when the initial datum of the shear flow satisfies $\left \| U(y)-y\right \|_{H^{N+6}}\ll 1$ , with $N>1$ , and the initial perturbations ${u}_{\mathrm {in}}$ and ${b}_{\mathrm {in}}$ satisfy $\left \| ( {u}_{\mathrm {in}},{b}_{\mathrm {in}}) \right \| _{H^{N+1}}=\epsilon \ll \nu ^{\frac 56+\tilde \delta }$ for any fixed $\tilde \delta >0$ , then the solution of the 2D MHD equations remains $\nu ^{-(\frac {1}{3}+\frac {\tilde \delta }{2})}\epsilon$ -close to $( e^{\nu t \partial _{yy}}U(y),0)^\top$ for all $t>0$ .

Refined conserved quantities criteria for the ideal MHD equations in a bounded domain

In this paper, we present two sufficient conditions keeping the total energy and cross-helicity conservation in spaces B _ 3 , V M O α ( Ω ) \underline {B}^{\alpha }_{3,VMO}(\Omega ) and B 3 , ∞ β ( Ω ) B^{\beta }_{3,\infty }(\Omega ) for the ideal magnetohydrodynamic equations in a bounded domain. This generalizes the sufficient criteria for conserved quantities recently obtained by Bardos, Gwiazda, Świerczewska Gwiazda, Titi and Wiedemann [Proc. A. 475 (2019), 18 pp.] and by Wang and Zuo [J. Differential Equations 268 (2020), pp. 4079–4101].

Optimal error analysis of an unconditionally stable BDF2 finite element approximation for the 3D incompressible MHD equations with variable density

This paper presents a second-order finite element scheme for the approximations of the three-dimensional (3D) incompressible magnetohydrodynamics system with variable density (VD-MHD), where two-step backward differentiation formula (BDF2) is used to discrete the time derivative and the (P2,P1b,P1,P1) finite elements are used to approximate the density, velocity, pressure and magnetic. Based on an equivalent VD-MHD system, the proposed finite element algorithm is unconditionally stable in the sense that the discrete energy inequalities hold without small condition of the time step size. After a rigorous analysis, the optimal L2 error estimates (τ2+h2) of the density, velocity field and magnetic field are established, where τ and h are the time step size and mesh size. Finally, numerical results are given to support these convergence rates.

Uniform attractors for 3D MHD equations with nonlinear damping

ABSTRACT The aim of this paper is to investigate the uniform attractors for 3D MHD equations with nonlinear damping. Some uniform estimates for strong solutions are established. Furthermore, we prove that 3D damped equations have an ( V , V ) -uniform attractor and an ( V , H 2 ) -uniform attractor, and the ( V , V ) -uniform attractor is actually the ( V , H 2 ) -uniform attractor.

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

Local well-posedness to the full compressible MHD equations without initial compatibility conditions

Local well-posedness to the full compressible MHD equations without initial compatibility conditions

Convergence Analysis of Some Finite Element Parallel Algorithms for the Stationary Incompressible MHD Equations

Convergence Analysis of Some Finite Element Parallel Algorithms for the Stationary Incompressible MHD Equations

Existence and Approximation of Statistical Solutions of the 3D MHD Equations

Existence and Approximation of Statistical Solutions of the 3D MHD Equations

三维不可压MHD方程在变指数Lebesgue空间中的适定性

三维不可压MHD方程在变指数Lebesgue空间中的适定性

Asymptotic Behavior of Solutions for the Three-dimensional Generalized Incompressible MHD Equations with Nonlinear Damping Terms

Asymptotic Behavior of Solutions for the Three-dimensional Generalized Incompressible MHD Equations with Nonlinear Damping Terms