294 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
A new thermodynamically compatible finite volume scheme for magnetohydrodynamics
In this paper we propose a novel thermodynamically compatible finite volume
scheme for the numerical solution of the equations of magnetohydrodynamics
(MHD) in one and two space dimensions. As shown by Godunov in 1972, the MHD
system can be written as overdetermined symmetric hyperbolic and
thermodynamically compatible (SHTC) system. More precisely, the MHD equations
are symmetric hyperbolic in the sense of Friedrichs and satisfy the first and
second principles of thermodynamics. In a more recent work on SHTC systems,
\cite{Rom1998}, the entropy density is a primary evolution variable, and total
energy conservation can be shown to be a \textit{consequence} that is obtained
after a judicious linear combination of all other evolution equations. The
objective of this paper is to mimic the SHTC framework also on the discrete
level by directly discretizing the \textit{entropy inequality}, instead of the
total energy conservation law, while total energy conservation is obtained via
an appropriate linear combination as a \textit{consequence} of the
thermodynamically compatible discretization of all other evolution equations.
As such, the proposed finite volume scheme satisfies a discrete cell entropy
inequality \textit{by construction} and can be proven to be nonlinearly stable
in the energy norm due to the discrete energy conservation. In multiple space
dimensions the divergence-free condition of the magnetic field is taken into
account via a new thermodynamically compatible generalized Lagrangian
multiplier (GLM) divergence cleaning approach. The fundamental properties of
the scheme proposed in this paper are mathematically rigorously proven. The new
method is applied to some standard MHD benchmark problems in one and two space
dimensions, obtaining good results in all cases
A Flux-Differencing Formula for Split-Form Summation By Parts Discretizations of Non-Conservative Systems: Applications to Subcell Limiting for magneto-hydrodynamics
In this paper, we show that diagonal-norm summation by parts (SBP)
discretizations of general non-conservative systems of hyperbolic balance laws
can be rewritten as a finite-volume-type formula, also known as
flux-differencing formula, if the non-conservative terms can be written as the
product of a local and a symmetric contribution. Furthermore, we show that the
existence of a flux-differencing formula enables the use of recent subcell
limiting strategies to improve the robustness of the high-order
discretizations.
To demonstrate the utility of the novel flux-differencing formula, we
construct hybrid schemes that combine high-order SBP methods (the discontinuous
Galerkin spectral element method and a high-order SBP finite difference method)
with a compatible low-order finite volume (FV) scheme at the subcell level. We
apply the hybrid schemes to solve challenging magnetohydrodynamics (MHD)
problems featuring strong shocks
Lagrangian ADER-WENO Finite Volume Schemes on Unstructured Triangular Meshes Based On Genuinely Multidimensional HLL Riemann Solvers
In this paper we use the genuinely multidimensional HLL Riemann solvers
recently developed by Balsara et al. to construct a new class of
computationally efficient high order Lagrangian ADER-WENO one-step ALE finite
volume schemes on unstructured triangular meshes. A nonlinear WENO
reconstruction operator allows the algorithm to achieve high order of accuracy
in space, while high order of accuracy in time is obtained by the use of an
ADER time-stepping technique based on a local space-time Galerkin predictor.
The multidimensional HLL and HLLC Riemann solvers operate at each vertex of the
grid, considering the entire Voronoi neighborhood of each node and allows for
larger time steps than conventional one-dimensional Riemann solvers. The
results produced by the multidimensional Riemann solver are then used twice in
our one-step ALE algorithm: first, as a node solver that assigns a unique
velocity vector to each vertex, in order to preserve the continuity of the
computational mesh; second, as a building block for genuinely multidimensional
numerical flux evaluation that allows the scheme to run with larger time steps
compared to conventional finite volume schemes that use classical
one-dimensional Riemann solvers in normal direction. A rezoning step may be
necessary in order to overcome element overlapping or crossing-over. We apply
the method presented in this article to two systems of hyperbolic conservation
laws, namely the Euler equations of compressible gas dynamics and the equations
of ideal classical magneto-hydrodynamics (MHD). Convergence studies up to
fourth order of accuracy in space and time have been carried out. Several
numerical test problems have been solved to validate the new approach
Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows
In this work, we consider the discretization of nonlinear hyperbolic systems
in nonconservative form with the high-order discontinuous Galerkin spectral
element method (DGSEM) based on collocation of quadrature and interpolation
points (Kopriva and Gassner, J. Sci. Comput., 44 (2010), pp.136--155; Carpenter
et al., SIAM J. Sci. Comput., 36 (2014), pp.~B835-B867). We present a general
framework for the design of such schemes that satisfy a semi-discrete entropy
inequality for a given convex entropy function at any approximation order. The
framework is closely related to the one introduced for conservation laws by
Chen and Shu (J. Comput. Phys., 345 (2017), pp.~427--461) and relies on the
modification of the integral over discretization elements where we replace the
physical fluxes by entropy conservative numerical fluxes from Castro et al.
(SIAM J. Numer. Anal., 51 (2013), pp.~1371--1391), while entropy stable
numerical fluxes are used at element interfaces. Time discretization is
performed with strong-stability preserving Runge-Kutta schemes. We use this
framework for the discretization of two systems in one space-dimension: a
system with a nonconservative product associated to a
linearly-degenerate field for which the DGSEM fails to capture the physically
relevant solution, and the isentropic Baer-Nunziato model. For the latter, we
derive conditions on the numerical parameters of the discrete scheme to further
keep positivity of the partial densities and a maximum principle on the void
fractions. Numerical experiments support the conclusions of the present
analysis and highlight stability and robustness of the present schemes
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