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## A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/09/2015

Relevância na Pesquisa

36.5%

We propose a high-order finite difference weighted ENO (WENO) method for the
ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage
(i.e. it has no internal stages to store), single-step (i.e. it has no time
history that needs to be stored), maintains a discrete divergence-free
condition on the magnetic field, and has the capacity to preserve the
positivity of the density and pressure. To accomplish this, we use a Taylor
discretization of the Picard integral formulation (PIF) of the finite
difference WENO method proposed in [SINUM, 53 (2015), pp. 1833-1856], where the
focus is on a high-order discretization of the fluxes (as opposed to the
conserved variables). We use constrained transport in order to obtain
divergence-free magnetic fields, which means that we simultaneously evolve the
magnetohydrodynamic (which has an evolution equation for the magnetic field)
and magnetic potential equations alongside each other, and set the magnetic
field to be the (discrete) curl of the magnetic potential after each time step.
In order to retain a single-stage, single-step method, we develop a novel
Lax-Wendroff discretization for the evolution of the magnetic potential, where
we start with technology used for Hamilton-Jacobi equations in order to
construct a non-oscillatory magnetic field. Positivity preservation is realized
by introducing a parameterized flux limiter that considers a linear combination
of the high and low-order numerical fluxes. The choice of the free parameter is
then given in such a way that the fluxes are limited towards the low-order
solver until positivity is attained. We present two and three dimensional
numerical results for several standard test problems including a smooth Alfven
wave...

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## Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.5%

In this work we develop a class of high-order finite difference weighted
essentially non-oscillatory (FD-WENO) schemes for solving the ideal
magnetohydrodynamic (MHD) equations in 2D and 3D. The philosophy of this work
is to use efficient high-order WENO spatial discretizations with high-order
strong stability-preserving Runge-Kutta (SSP-RK) time-stepping schemes.
Numerical results have shown that with such methods we are able to resolve
solution structures that are only visible at much higher grid resolutions with
lower-order schemes. The key challenge in applying such methods to ideal MHD is
to control divergence errors in the magnetic field. We achieve this by
augmenting the base scheme with a novel high-order constrained transport
approach that updates the magnetic vector potential. The predicted magnetic
field from the base scheme is replaced by a divergence-free magnetic field that
is obtained from the curl of this magnetic potential. The non-conservative
weakly hyperbolic system that the magnetic vector potential satisfies is solved
using a version of FD-WENO developed for Hamilton-Jacobi equations. The
resulting numerical method is endowed with several important properties: (1)
all quantities, including all components of the magnetic field and magnetic
potential...

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## Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.5%

In this paper, we utilize the maximum-principle-preserving flux limiting
technique, originally designed for high order weighted essentially
non-oscillatory (WENO) methods for scalar hyperbolic conservation laws, to
develop a class of high order positivity-preserving finite difference WENO
methods for the ideal magnetohydrodynamic (MHD) equations. Our schemes, under
the constrained transport (CT) framework, can achieve high order accuracy, a
discrete divergence-free condition and positivity of the numerical solution
simultaneously. Numerical examples in 1D, 2D and 3D are provided to demonstrate
the performance of the proposed method.; Comment: 21 pages, 28 figures

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