22 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
The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement
We present the first implementation of the Active Flux method on adaptively
refined Cartesian grids. The Active Flux method is a third order accurate
finite volume method for hyperbolic conservation laws, which is based on the
use of point values as well as cell average values of the conserved quantities.
The resulting method has a compact stencil in space and time and good stability
properties.
The method is implemented as a new solver in ForestClaw, a software for
parallel adaptive mesh refinement of patch-based solvers. On each Cartesian
grid patch the single grid Active Flux method can be applied. The exchange of
data between grid patches is organised via ghost cells. The local stencil in
space and time and the availability of the point values that are used for the
reconstruction, leads to an efficient implementation. The resulting method is
third order accurate, conservative and allows the use of subcycling in time