4,507 research outputs found
Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics
In this work, we analyze a recently proposed stabilized finite element formulation for the approximation of the resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing ones is the fact that it always converges to the physical solution, even when it is singular. We have performed a detailed stability and convergence analysis of the formulation in a simplified setting. From the convergence analysis, we infer that a particular type of meshes with a macro-element structure is needed, which can be easily obtained after a straight modification of any original mesh.Preprin
Entropy Stable Finite Volume Approximations for Ideal Magnetohydrodynamics
This article serves as a summary outlining the mathematical entropy analysis
of the ideal magnetohydrodynamic (MHD) equations. We select the ideal MHD
equations as they are particularly useful for mathematically modeling a wide
variety of magnetized fluids. In order to be self-contained we first motivate
the physical properties of a magnetic fluid and how it should behave under the
laws of thermodynamics. Next, we introduce a mathematical model built from
hyperbolic partial differential equations (PDEs) that translate physical laws
into mathematical equations. After an overview of the continuous analysis, we
thoroughly describe the derivation of a numerical approximation of the ideal
MHD system that remains consistent to the continuous thermodynamic principles.
The derivation of the method and the theorems contained within serve as the
bulk of the review article. We demonstrate that the derived numerical
approximation retains the correct entropic properties of the continuous model
and show its applicability to a variety of standard numerical test cases for
MHD schemes. We close with our conclusions and a brief discussion on future
work in the area of entropy consistent numerical methods and the modeling of
plasmas
On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics
In this work, we propose a new stabilized finite element formulation for the approximation of the
resistive magnetohydrodynamics equations. The novelty of this formulation with respect to existing
ones is the fact that it always converges to the physical solution, even for singular ones. We have performed
a detailed stability and convergence analysis of the formulation in a simplified setting. From
the convergence analysis, we infer that a particular type of meshes with a macro-element structure is
needed, which can be easily obtained after a straight modification of any original mesh. A detailed
set of numerical experiments have been performed in order to validate our approach.Peer ReviewedPreprin
Affordable, Entropy Conserving and Entropy Stable Flux Functions for the Ideal MHD Equations
In this work, we design an entropy stable, finite volume approximation for
the ideal magnetohydrodynamics (MHD) equations. The method is novel as we
design an affordable analytical expression of the numerical interface flux
function that discretely preserves the entropy of the system. To guarantee the
discrete conservation of entropy requires the addition of a particular source
term to the ideal MHD system. Exact entropy conserving schemes cannot dissipate
energy at shocks, thus to compute accurate solutions to problems that may
develop shocks, we determine a dissipation term to guarantee entropy stability
for the numerical scheme. Numerical tests are performed to demonstrate the
theoretical findings of entropy conservation and robustness.Comment: arXiv admin note: substantial text overlap with arXiv:1509.06902;
text overlap with arXiv:1007.2606 by other author
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