42 research outputs found
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
Positivity-Preserving Finite Difference WENO Schemes with Constrained Transport for Ideal Magnetohydrodynamic Equations
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 figure