6,162 research outputs found
Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations
Two-fluid ideal plasma equations are a generalized form of the ideal MHD equations in which electrons and ions are considered as separate species. The design of efficient numerical schemes for the these equations is complicated on account of their non-linear nature and the presence of stiff source terms, especially for high charge to mass ratios and for low Larmor radii. In this article, we design entropy stable finite difference schemes for the two-fluid equations by combining entropy conservative fluxes and suitable numerical diffusion operators. Furthermore, to overcome the time step restrictions imposed by the stiff source terms, we devise time-stepping routines based on implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the two-fluid plasma equations is exploited by us to design IMEX schemes in which only local (in each cell) linear equations need to be solved at each time step. Benchmark numerical experiments are presented to illustrate the robustness and accuracy of these scheme
Entropy Stable Numerical Schemes for Two-Fluid Plasma Equations
Two-fluid ideal plasma equations are a generalized form of the ideal MHD
equations in which electrons and ions are considered as separate species. The
design of efficient numerical schemes for the these equations is complicated on
account of their non-linear nature and the presence of stiff source terms,
especially for high charge to mass ratios and for low Larmor radii. In this
article, we design entropy stable finite difference schemes for the two-fluid
equations by combining entropy conservative fluxes and suitable numerical
diffusion operators. Furthermore, to overcome the time step restrictions
imposed by the stiff source terms, we devise time-stepping routines based on
implicit-explicit (IMEX)-Runge Kutta (RK) schemes. The special structure of the
two-fluid plasma equations is exploited by us to design IMEX schemes in which
only local (in each cell) linear equations need to be solved at each time step.
Benchmark numerical experiments are presented to illustrate the robustness and
accuracy of these schemes.Comment: Accepted in Journal of Scientific Computin
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
An Unstaggered Constrained Transport Method for the 3D Ideal Magnetohydrodynamic Equations
Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations
in more than one space dimension must either confront the challenge of
controlling errors in the discrete divergence of the magnetic field, or else be
faced with nonlinear numerical instabilities. One approach for controlling the
discrete divergence is through a so-called constrained transport method, which
is based on first predicting a magnetic field through a standard finite volume
solver, and then correcting this field through the appropriate use of a
magnetic vector potential. In this work we develop a constrained transport
method for the 3D ideal MHD equations that is based on a high-resolution wave
propagation scheme. Our proposed scheme is the 3D extension of the 2D scheme
developed by Rossmanith [SIAM J. Sci. Comp. 28, 1766 (2006)], and is based on
the high-resolution wave propagation method of Langseth and LeVeque [J. Comp.
Phys. 165, 126 (2000)]. In particular, in our extension we take great care to
maintain the three most important properties of the 2D scheme: (1) all
quantities, including all components of the magnetic field and magnetic
potential, are treated as cell-centered; (2) we develop a high-resolution wave
propagation scheme for evolving the magnetic potential; and (3) we develop a
wave limiting approach that is applied during the vector potential evolution,
which controls unphysical oscillations in the magnetic field. One of the key
numerical difficulties that is novel to 3D is that the transport equation that
must be solved for the magnetic vector potential is only weakly hyperbolic. In
presenting our numerical algorithm we describe how to numerically handle this
problem of weak hyperbolicity, as well as how to choose an appropriate gauge
condition. The resulting scheme is applied to several numerical test cases.Comment: 46 pages, 12 figure
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
Ehrenfest regularization of Hamiltonian systems
Imagine a freely rotating rigid body. The body has three principal axes of
rotation. It follows from mathematical analysis of the evolution equations that
pure rotations around the major and minor axes are stable while rotation around
the middle axis is unstable. However, only rotation around the major axis (with
highest moment of inertia) is stable in physical reality (as demonstrated by
the unexpected change of rotation of the Explorer 1 probe). We propose a
general method of Ehrenfest regularization of Hamiltonian equations by which
the reversible Hamiltonian equations are equipped with irreversible terms
constructed from the Hamiltonian dynamics itself. The method is demonstrated on
harmonic oscillator, rigid body motion (solving the problem of stable minor
axis rotation), ideal fluid mechanics and kinetic theory. In particular, the
regularization can be seen as a birth of irreversibility and dissipation. In
addition, we discuss and propose discretizations of the Ehrenfest regularized
evolution equations such that key model characteristics (behavior of energy and
entropy) are valid in the numerical scheme as well
A multidimensional grid-adaptive relativistic magnetofluid code
A robust second order, shock-capturing numerical scheme for multi-dimensional
special relativistic magnetohydrodynamics on computational domains with
adaptive mesh refinement is presented. The base solver is a total variation
diminishing Lax-Friedrichs scheme in a finite volume setting and is combined
with a diffusive approach for controlling magnetic monopole errors. The
consistency between the primitive and conservative variables is ensured at all
limited reconstructions and the spatial part of the four velocity is used as a
primitive variable. Demonstrative relativistic examples are shown to validate
the implementation. We recover known exact solutions to relativistic MHD
Riemann problems, and simulate the shock-dominated long term evolution of
Lorentz factor 7 vortical flows distorting magnetic island chains.Comment: accepted for publication in Computer Physics Communication
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