Resolution of the system of equations of ideal magnetohydrodynamics by a finite volume kinetic-type method

The numerical investigation of a finite volume scheme for the ideal MHD, introduced by the authors in a preceding paper, is pursued. This scheme is based on an original flux-splitting formula, partly kinetic based. It is simple, robust and is inherently consistent with the equation ∇ · B = 0 on the magnetic field. Numerical tests in one and two dimensions with the second order version of the scheme are displayed, showing its efficiency. The numerical dissipation is estimated.     

Upwind methods for hyperbolic conservation laws with source terms

This paper deals with the extension of some upwind schemes to hyperbolic systems of conservation laws with source term. More precisely we give methods to get natural upwind discretizations of the source term when the flux is approximated by using flux-difference or flux-splitting techniques. In particular, the Q-schemes of Roe and van Leer and the flux-splitting techniques of Steger-Warming and Vijayasundaram are considered. Numerical results for a scalar advection equation with nonlinear source and for the one-dimensional shallow water equations are presented. In the last case we compare the different schemes proposed in terms of a conservation property. When this property does not hold, spurious numerical waves can appear which is the case for the centred discretization of the source term.     

Mathematical modeling of the MHD pump in view of the external circuit

This article presents spatially a 2D cylindrically symmetrical, mathematical model of the MHD pump that is not stationary in time. To describe electromagnetic fields, the MHD approximation of Maxwell’s equations is used; to describe the motion of heat-transfer fluid in the pump channel, the Navier-Stokes equations written in terms of vortex and flow function are used. The fluid consumption is calculated from the equation of fluid motion in a closed circuit. The calculations of different pump operating modes are presented.     

Magnetohydrodynamic state estimation with boundary sensors

We present a PDE observer that estimates the velocity, pressure, electric potential and current fields in a magnetohydrodynamic (MHD) channel flow, also known as Hartmann flow. This flow is characterized by an electrically conducting fluid moving between parallel plates in the presence of an externally imposed transverse magnetic field. The system is described by the inductionless MHD equations, a combination of the Navier-Stokes equations and a Poisson equation for the electric potential under the so-called inductionless MHD approximation in a low magnetic Reynolds number regime. We identify physical quantities (measurable on the wall of the channel) that are sufficient to generate convergent estimates of the velocity, pressure, and electric potential field away from the walls. Our observer consists of a copy of the linearized MHD equations, combined with linear injection of output estimation error, with observer gains designed using backstepping. Pressure, skin friction and current measurements from one of the walls are used for output injection. For zero magnetic field or nonconducting fluid, the design reduces to an observer for the Navier-Stokes Poiseuille flow, a benchmark for flow control and turbulence estimation. We show that for the linearized MHD model the estimation error converges to zero in the L² norm. Despite being a subject of practical interest, the problem of observer design for nondiscretized 3-D MHD or Navier-Stokes channel flow has so far been an open problem.     

Lattice kinetic simulations of 3-D MHD turbulence

A recently proposed lattice Boltzmann kinetic scheme offers a promising tool for simulating complex 3-D MHD flows. The algorithm is based on the BGK modeling of the collision term. The conventional approach for implementing magnetic behavior in LBM methods is based on one tensor-valued distribution function to present both the fluids variables (density and momentum) and the magnetic field. This formulation, however, has been proven a rather inefficient approach. The present scheme calls for a separate BGK-like evolution equation for the magnetic field which models the induction equation and enhances simplicity while allowing for the independent adjustment of the magnetic resistivity. Furthermore the algorithm correctly recovers the macroscopic dissipative MHD equations. Numerical results for the 3-D Taylor–Green vortex problem are presented with corresponding results computed with a pseudo-spectral code used as benchmark.     

Implementation of the CIP algorithm to magnetohydrodynamic simulations

An implementation of the Constrained Interpolation Profile (CIP) algorithm to magnetohydrodynamic (MHD) simulations is presented. First we transform the original momentum and magnetic induction equations to unfamiliar forms by introducing Elsässer variables [W.M. Elsässer, The hydromagnetic equations, Phys. Rev. (1950)]. In this formulation, while the compressional and pressure gradient terms remain as non-advective terms, the advective and magnetic stress terms are expressed in the form of an advection equation, which enables us to use the CIP algorithm. We have examined some 1D test problems using the code based on this formula. Linear Alfvén wave propagation tests reveal that the developed code is capable of solving any Alfvén wave propagation with only small numerical diffusion and phase errors up to k?h=2.5 (where ?h is the grid spacing). A shock tube test shows good agreement with a previous result with less numerical oscillation at the shock front and the contact discontinuity which are captured within a few grid points. Extension of the 1D code to the multi-dimensional case is straightforward. We have calculated the 3D nonlinear evolution of the Kelvin-Helmholtz instability (KHI) and compared the result with our previous study. We find that our new MHD code is capable of following the 3D turbulence excited by the KHI while retaining the solenoidal property of the magnetic field.     

A new MHD code with adaptive mesh refinement and parallelization for astrophysics

A new code, named MAP, is written in FORTRAN language for magnetohydrodynamics (MHD) simulations with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are several optional numerical schemes for computing the MHD part, namely, modified Mac Cormack Scheme (MMC), Lax–Friedrichs scheme (LF), and weighted essentially non-oscillatory (WENO) scheme. All of them are second-order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation diminishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh. We use the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the code. The details of the AMR and MPI algorithms are described in the paper.     

Globally Divergence-Free Discontinuous Galerkin Methods for Ideal Magnetohydrodynamic Equations

Ideal magnetohydrodynamic (MHD) equations are widely used in many areas in physics and engineering, and these equations have a divergence-free constraint on the magnetic field. In this paper, we propose high order globally divergence-free numerical methods to solve the ideal MHD equations. The algorithms are based on discontinuous Galerkin methods in space. The induction equation is discretized separately to approximate the normal components of the magnetic field on elements interfaces, and to extract additional information about the magnetic field when higher order accuracy is desired. This is then followed by an element by element reconstruction to obtain the globally divergence-free magnetic field. In time, strong-stability-preserving Runge–Kutta methods are applied. In consideration of accuracy and stability of the methods, a careful investigation is carried out, both numerically and analytically, to study the choices of the numerical fluxes associated with the electric field at element interfaces and vertices. The resulting methods are local and the approximated magnetic fields are globally divergence-free. Numerical examples are presented to demonstrate the accuracy and robustness of the methods.     

Chebyshev wavelet collocation method for magnetohydrodynamic flow equations

This study proposes Chebyshev wavelet collocation method for partial differential equation and applies to solve magnetohydrodynamic (MHD) flow equations in a rectangular duct in the presence of transverse external oblique magnetic field. Approximate solutions of velocity and induced magnetic field are obtained for steady‐state, fully developed, incompressible flow for a conducting fluid inside the duct. Numerical results of the MHD flow problem show that the accuracy of proposed method is quite good even in the case of a small number of grid points. The results for velocity and induced magnetic field are visualized in terms of graphics for values of Hartmann number H_a ≤ 1000.

     

A subgrid stabilization finite element method for incompressible magnetohydrodynamics

This paper studies a numerical scheme for approximating solutions of incompressible magnetohydrodynamic (MHD) equations that uses eddy viscosity stabilization only on the small scales of the fluid flow. This stabilization scheme for MHD equations uses a Galerkin finite element spatial discretization with Scott-Vogelius mixed finite elements and semi-implicit backward Euler temporal discretization. We prove its unconditional stability and prove how the coarse mesh can be chosen so that optimal convergence can be achieved. We also provide numerical experiments to confirm the theory and demonstrate the effectiveness of the scheme on a test problem for MHD channel flow.     

Meshless local boundary integral equation (LBIE) method for the unsteady magnetohydrodynamic (MHD) flow in rectangular and circular pipes

The meshless local boundary integral equation (LBIE) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady magnetohydrodynamic (MHD) flow through a pipe of rectangular and circular sections with non-conducting walls. Computations have been carried out for different Hartmann numbers and at various time levels. The method is based on the local boundary integral equation with moving least squares (MLS) approximation. For the MLS, nodal points spread over the analyzed domain, are utilized to approximate the interior and boundary variables. A time stepping method is employed to deal with the time derivative. Finally, numerical results are presented to show the behaviour of velocity and induced magnetic field.     

High β and toroidal effects in three dimensions

The resistive MHD equations for toroidal plasma configurations are reduced by expanding to second order in ?, the inverse aspect ratio, allowing for high β = μ₀p/B² of order ?. The result is a closed system of nonlinear, three-dimensional equations where the fast magnetohydrodynamic time scale is eliminated. In particular, the equation for the toroidal velocity remains decoupled. These equations generalize reduced equations derived earlier. They are now solved numerically. As a first step, various types of axisymmetric equilibria are found by relaxing a given initial configuration. Then, three-dimensional perturbations are introduced and followed in time to investigate the linear and nonlinear properties of tearing modes, etc., for high-β plasmas in toroidal geometry.     

A coarse-grid projection method for accelerating incompressible MHD flow simulations

Coarse grid projection (CGP) is a multiresolution technique for accelerating numerical calculations associated with a set of nonlinear evolutionary equations along with stiff Poisson’s equations. In this article, we use CGP for the first time to speed up incompressible magnetohydrodynamics (MHD) flow simulations. Accordingly, we solve the nonlinear advection–diffusion equation on a fine mesh, while we execute the electric potential Poisson equation on the corresponding coarsened mesh. Mapping operators connect two grids together. A pressure correction scheme is used to enforce the incompressibility constrain. The study of incompressible flow past a circular cylinder in the presence of Lorentz force is selected as a benchmark problem with a fixed Reynolds number but various Stuart numbers. We consider two different situations. First, we only apply CGP to the electric potential Poisson equation. Second, we apply CGP to the pressure Poisson equation as well. The maximum speed-up factors achieved here are approximately 3 and 23, respectively, for the first and second situations. For the both situations, we examine the accuracy of velocity and vorticity fields as well as the lift and drag coefficients. In general, the results obtained by CGP are in an excellent to reasonable range of accuracy. The CGP results are significantly more accurate compared to the numerical simulations of the advection–diffusion and electric potential Poisson equations on pure coarse scale grids.

     

Solution formulas for cubic equations without or with constraints

We present a convention (for square/cubic root) which provides correct interpretations of the Lagrange formula for all cubic polynomial equations with real coefficients. Using this convention, we also present a real solution formula for the general cubic equation with real coefficients under equality and inequality constraints.     

A generalized solution of an initial-boundary value problem arising in the mechanics of discrete-continuous systems

We define a generalized solution of an initial-boundary value problem for a linear system of differential equations with one ordinary differential equation and two partial differential equations (a hybrid system of differential equations). We prove that the problem is well-posed and has a unique generalized solution. An analytical formula for the solution is found. Such systems of differential equations arise in studying discrete-continuum mechanical systems.     

The modified differential transform method for solving MHD boundary-layer equations

A new analytical method (DTM-Padé) was developed for solving magnetohydrodynamic boundary-layer equations. It was shown that differential transform method (DTM) solutions are only valid for small values of independent variable. Therefore the DTM is not applicable for solving MHD boundary-layer equations, because in the boundary-layer problem y→∞. Numerical comparisons between the DTM-Padé and numerical methods (by using a fourth-order Runge-Kutta and shooting method) revealed that the new technique is a powerful method for solving MHD boundary-layer equations.     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

MINERVA: Ideal MHD stability code for toroidally rotating tokamak plasmas

A new linear MHD stability code MINERVA is developed for investigating a toroidal rotation effect on the stability of ideal MHD modes in tokamak plasmas. This code solves the Frieman-Rotenberg equation as not only the generalized eigenvalue problem but also the initial value problem. The parallel computing method used in this code realizes the stability analysis of both long and short wavelength MHD modes in short time. The results of some benchmarking tests show the validity of this MINERVA code. The numerical study with MINERVA about the toroidal rotation effect on the edge MHD stability shows that the rotation shear destabilizes the intermediate wavelength modes but stabilizes the short wavelength edge localized MHD modes, though the rotation frequency destabilizes both the long and the short wavelength MHD modes.     

Taylor wavelet method for fractional delay differential equations

We present a new numerical method for solving fractional delay differential equations. The method is based on Taylor wavelets. We establish an exact formula to determine the Riemann–Liouville fractional integral of the Taylor wavelets. The exact formula is then applied to reduce the problem of solving a fractional delay differential equation to the problem of solving a system of algebraic equations. Several numerical examples are presented to show the applicability and the effectiveness of this method.

     

Numerical model of a two-dimensional,non-plane transient magnetohydrodynamic flow

The equations describing two-dimensional three-component magnetohydrodynamic (MHD) transient flows are formulated for a system of spherical coordinates. With the numerical code based on Implicit Continuous Fluid Eulerian (ICE) scheme, MHD flows resulting from a sudden energy release in a stratified medium are examined. Because of the inclusion of out-of-plane components of velocity and magnetic fields, MHD transverse waves are observed in addition to fast, slow and entropy waves. Numerical results for compressible MHD shocks are found in satisfactory agreement with the theoretical predictions.