3,923 research outputs found
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
Very High Order \PNM Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations
In this paper we propose the first better than second order accurate method
in space and time for the numerical solution of the resistive relativistic
magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space
dimensions. The nonlinear system under consideration is purely hyperbolic and
contains a source term, the one for the evolution of the electric field, that
becomes stiff for low values of the resistivity. For the spatial discretization
we propose to use high order \PNM schemes as introduced in \cite{Dumbser2008}
for hyperbolic conservation laws and a high order accurate unsplit time
discretization is achieved using the element-local space-time discontinuous
Galerkin approach proposed in \cite{DumbserEnauxToro} for one-dimensional
balance laws with stiff source terms. The divergence free character of the
magnetic field is accounted for through the divergence cleaning procedure of
Dedner et al. \cite{Dedneretal}. To validate our high order method we first
solve some numerical test cases for which exact analytical reference solutions
are known and we also show numerical convergence studies in the stiff limit of
the RRMHD equations using \PNM schemes from third to fifth order of accuracy
in space and time. We also present some applications with shock waves such as a
classical shock tube problem with different values for the conductivity as well
as a relativistic MHD rotor problem and the relativistic equivalent of the
Orszag-Tang vortex problem. We have verified that the proposed method can
handle equally well the resistive regime and the stiff limit of ideal
relativistic MHD. For these reasons it provides a powerful tool for
relativistic astrophysical simulations involving the appearance of magnetic
reconnection.Comment: 24 pages, 6 figures, submitted to JC
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
A Constrained Transport Method for the Solution of the Resistive Relativistic MHD Equations
We describe a novel Godunov-type numerical method for solving the equations
of resistive relativistic magnetohydrodynamics. In the proposed approach, the
spatial components of both magnetic and electric fields are located at zone
interfaces and are evolved using the constrained transport formalism. Direct
application of Stokes' theorem to Faraday's and Ampere's laws ensures that the
resulting discretization is divergence-free for the magnetic field and
charge-conserving for the electric field. Hydrodynamic variables retain,
instead, the usual zone-centred representation commonly adopted in
finite-volume schemes. Temporal discretization is based on Runge-Kutta
implicit-explicit (IMEX) schemes in order to resolve the temporal scale
disparity introduced by the stiff source term in Ampere's law. The implicit
step is accomplished by means of an improved and more efficient Newton-Broyden
multidimensional root-finding algorithm. The explicit step relies on a
multidimensional Riemann solver to compute the line-averaged electric and
magnetic fields at zone edges and it employs a one-dimensional Riemann solver
at zone interfaces to update zone-centred hydrodynamic quantities. For the
latter, we introduce a five-wave solver based on the frozen limit of the
relaxation system whereby the solution to the Riemann problem can be decomposed
into an outer Maxwell solver and an inner hydrodynamic solver. A number of
numerical benchmarks demonstrate that our method is superior in stability and
robustness to the more popular charge-conserving divergence cleaning approach
where both primary electric and magnetic fields are zone-centered. In addition,
the employment of a less diffusive Riemann solver noticeably improves the
accuracy of the computations.Comment: 25 pages, 14 figure
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
An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry
We design an arbitrary high-order accurate nodal discontinuous Galerkin
spectral element approximation for the nonlinear two dimensional shallow water
equations with non-constant, possibly discontinuous, bathymetry on
unstructured, possibly curved, quadrilateral meshes. The scheme is derived from
an equivalent flux differencing formulation of the split form of the equations.
We prove that this discretisation exactly preserves the local mass and
momentum. Furthermore, combined with a special numerical interface flux
function, the method exactly preserves the mathematical entropy, which is the
total energy for the shallow water equations. By adding a specific form of
interface dissipation to the baseline entropy conserving scheme we create a
provably entropy stable scheme. That is, the numerical scheme discretely
satisfies the second law of thermodynamics. Finally, with a particular
discretisation of the bathymetry source term we prove that the numerical
approximation is well-balanced. We provide numerical examples that verify the
theoretical findings and furthermore provide an application of the scheme for a
partial break of a curved dam test problem
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