Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Kuzmin, Dmitri; Klyushnev, N. V.

doi:10.1016/j.jcp.2020.109230

Cited by 15 publications

(3 citation statements)

References 45 publications

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“…In [26], we proved that the FE approximation of residual-based artificial viscosity method applied to scalar conservation laws converges to the unique entropy solution, and we extended the method to solve more general systems in the framework of DG, spectral elements, finite differences, and radial basis functions (RBF), see e.g., [27][28][29][30][31]. For other approaches to constructing the nonlinear artificial viscosity methods, we refer to the work of [32][33][34], where the stabilization is constructed based on a smoothness indicator of the solution.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

Dao

Nazarov

2022

J Sci Comput

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We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

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Section: Introductionmentioning

confidence: 99%

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

Dao

Nazarov

2022

J Sci Comput

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“…Another problem is that unless the thermal diffusivity is zero, the resistive MHD flux violates the minimum entropy principle, see [11]. On the other hand, the non-physically-motivated monolithic viscous flux is employed in numerical schemes for MHD, e.g., [20,24] as well as being a continuous analog of the well-known Lax-Friedrichs or upwind schemes. However, to the best of our knowledge, investigations regarding entropy principles and other positivity-preserving properties of viscous regularizations to the MHD equations are still missing in the literature.…”

Section: Introductionmentioning

confidence: 99%

Monolithic parabolic regularization of the MHD equations and entropy principles

Dao,

Nazarov

2022

Preprint

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We show at the PDE level that the monolithic parabolic regularization of the equations of ideal magnetohydrodynamics (MHD) is compatible with all the generalized entropies, fulfills the minimum entropy principle, and preserves the positivity of density and internal energy. We then numerically investigate this regularization for the MHD equations using continuous finite elements in space and explicit strong stability preserving Runge-Kuta methods in time. The artificial viscosity coefficient of the regularization term is constructed to be proportional to the entropy residual of MHD. It is shown that the method has a high order of accuracy for smooth problems and captures strong shocks and discontinuities accurately for non-smooth problems.

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“…Efforts have been devoted to the development of divergence-free techniques. For example, in [39] the divergence equation ∇ • B = 0 is taken into account through a Lagrange multiplier that is additionally introduced in the set of the unknowns; in [50], the divergence-free condition relies on special flux limiters; in [46] a special energy functional is minimized by a least squares finite element method. Instead, the VEM considered in this chapter provides a numerical approximation of the magnetic field that is intrinsically divergence free as a consequence of a de Rham inequality chain.…”

Section: Introductionmentioning

confidence: 99%

The virtual element method for the coupled system of magneto-hydrodynamics

Naranjo-Alvarez¹,

Bokil²,

Gyrya³

et al. 2021

Preprint

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In this work, we review the framework of the Virtual Element Method (VEM) for a model in magneto-hydrodynamics (MHD), that incorporates a coupling between electromagnetics and fluid flow, and allows us to construct novel discretizations for simulating realistic phenomenon in MHD. First, we study two chains of spaces approximating the electromagnetic and fluid flow components of the model. Then, we show that this VEM approximation will yield divergence free discrete magnetic fields, an important property in any simulation in MHD. We present a linearization strategy to solve the VEM approximation which respects the divergence free condition on the magnetic field. This linearization will require that, at each non-linear iteration, a linear system be solved. We study these linear systems and show that they represent well-posed saddle point problems. We conclude by presenting numerical experiments exploring the performance of the VEM applied to the subsystem describing the electromagnetics. The first set of experiments provide evidence regarding the speed of convergence of the method as well as the divergence-free condition on the magnetic field. In the second set we present a model for magnetic reconnection in a mesh that includes a series of hanging nodes, which we use to calibrate the resolution of the method. The magnetic reconnection phenomenon happens near the center of the domain where the mesh resolution is finer and high resolution is achieved.

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Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Cited by 15 publications

References 45 publications

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

Monolithic parabolic regularization of the MHD equations and entropy principles

The virtual element method for the coupled system of magneto-hydrodynamics

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