46 research outputs found
High-Order Discontinuous Galerkin Finite Element Methods with Globally Divergence-Free Constrained Transport for Ideal MHD
The modification of the celebrated Yee scheme from Maxwell equations to
magnetohydrodynamics is often referred to as the constrained transport
approach. Constrained transport can be viewed as a sort of predictor-corrector
method for updating the magnetic field, where a magnetic field value is first
predicted by a method that does not preserve the divergence-free condition on
the magnetic field, followed by a correction step that aims to control these
divergence errors. This strategy has been successfully used in conjunction with
a variety of shock-capturing methods including WENO, central, and wave
propagation schemes. In this work we show how to extend the basic CT framework
to the discontinuous Galerkin finite element method on both 2D and 3D Cartesian
grids. We first review the entropy-stability theory for semi-discrete DG
discretizations of ideal MHD, which rigorously establishes the need for a
magnetic field that satisfies the following conditions: (1) the divergence of
the magnetic field is zero on each element, and (2) the normal components of
the magnetic field are continuous across element edges/faces. In order to
achieve such a globally divergence-free magnetic field, we introduce a novel CT
scheme that is based on two ingredients: (1) we introduce an element-centered
magnetic vector potential that is updated via a discontinuous Galerkin scheme
on the induction equation; and (2) we define a mapping that takes
element-centered magnetic field values and element-centered magnetic vector
potential values and creates on each edge /face a representation of the normal
component of the magnetic field; this representation is then mapped back to the
elements to create a globally divergence-free element-centered representation
of the magnetic field. For problems with shock waves, we make use of so-called
moment-based limiters to control oscillations in the conserved quantities.Comment: 26 pages, 6 figure
A Class of Quadrature-Based Moment-Closure Methods with Application to the Vlasov-Poisson-Fokker-Planck System in the High-Field Limit
Quadrature-based moment-closure methods are a class of approximations that
replace high-dimensional kinetic descriptions with lower-dimensional fluid
models. In this work we investigate some of the properties of a sub-class of
these methods based on bi-delta, bi-Gaussian, and bi-B-spline representations.
We develop a high-order discontinuous Galerkin (DG) scheme to solve the
resulting fluid systems. Finally, via this high-order DG scheme and Strang
operator splitting to handle the collision term, we simulate the fluid-closure
models in the context of the Vlasov-Poisson-Fokker-Planck system in the
high-field limit. We demonstrate numerically that the proposed scheme is
asymptotic-preserving in the high-field limit.Comment: 24 pages, 5 figure
Outflow Positivity Limiting for Hyperbolic Conservation Laws. Part I: Framework and Recipe
Numerical methods for hyperbolic conservation laws are needed that
efficiently mimic the constraints satisfied by exact solutions, including
material conservation and positivity, while also maintaining high-order
accuracy and numerical stability. Discontinuous Galerkin (DG) and WENO schemes
allow efficient high-order accuracy while maintaining conservation. Positivity
limiters developed by Zhang and Shu ensure a minimum time step for which
positivity of cell average quantities is maintained without sacrificing
conservation or formal accuracy; this is achieved by linearly damping the
deviation from the cell average just enough to enforce a cell positivity
condition that requires positivity at boundary nodes and strategically chosen
interior points.
We assume that the set of positive states is convex; it follows that
positivity is equivalent to scalar positivity of a collection of affine
functionals. Based on this observation, we generalize the method of Zhang and
Shu to a framework that we call outflow positivity limiting: First, enforce
positivity at boundary nodes. If wave speed desingularization is needed, cap
wave speeds at physically justified maxima by using remapped states to
calculate fluxes. Second, apply linear damping again to cap the boundary
average of all positivity functionals at the maximum possible (relative to the
cell average) for a scalar-valued representation positive in each mesh cell.
This be done by enforcing positivity of the retentional, an affine combination
of the cell average and the boundary average, in the same way that Zhang and
Shu would enforce positivity at a single point (and with similar computational
expense). Third, limit the time step so that cell outflow is less than the
initial cell content. This framework guarantees essentially the same
positivity-preserving time step as is guaranteed if positivity is enforced at
every point in the mesh cell.Comment: 32 pages, 15 figure
Ten-moment two-fluid plasma model agrees well with PIC/Vlasov in GEM problem
We simulate magnetic reconnection in the GEM problem using a two-fluid model
with 10 moments for the electron fluid as well as the proton fluid. We show
that use of 10 moments for electrons gives good qualitative agreement with the
the electron pressure tensor components in published kinetic simulations.Comment: 13th International Conference on Hyperbolic Problems, June 201
Simulation of Fast Magnetic Reconnection using a Two-Fluid Model of Collisionless Pair Plasma without Anomalous Resistivity
For the first time to our knowledge, we demonstrate fast magnetic
reconnection near a magnetic null point in a fluid model of collisionless pair
plasma without resorting to the contrivance of anomalous resistivity. In
particular, we demonstrate that fast reconnection occurs in an anisotropic
adiabatic two-fluid model of collisionless pair plasma with relaxation toward
isotropy for a broad range of isotropization rates. For very rapid
isotropization we see fast reconnection, but instabilities eventually arise
that cause numerical error and cast doubt on the simulated behavior
A high-order unstaggered constrained transport method for the 3D ideal magnetohydrodynamic equations based on the method of lines
Numerical methods for solving the ideal magnetohydrodynamic (MHD) equations
in more than one space dimension must confront the challenge of controlling
errors in the discrete divergence of the magnetic field. One approach that has
been shown successful in stabilizing MHD calculations are constrained transport
(CT) schemes. CT schemes can be viewed as predictor-corrector methods for
updating the magnetic field, where a magnetic field value is first predicted by
a method that does not exactly preserve the divergence-free condition on the
magnetic field, followed by a correction step that aims to control these
divergence errors. In Helzel et al. (2011) the authors presented an unstaggered
constrained transport method for the MHD equations on 3D Cartesian grids. In
this work we generalize the method of Helzel et al. (2011) in three important
ways: (1) we remove the need for operator splitting by switching to an
appropriate method of lines discretization and coupling this with a
non-conservative finite volume method for the magnetic vector potential
equation, (2) we increase the spatial and temporal order of accuracy of the
entire method to third order, and (3) we develop the method so that it is
applicable on both Cartesian and logically rectangular mapped grids. The
evolution equation for the magnetic vector potential is solved using a
non-conservative finite volume method. The curl of the magnetic potential is
computed via a third-order accurate discrete operator that is derived from
appropriate application of the divergence theorem and subsequent numerical
quadrature on element faces. Special artificial resistivity limiters are used
to control unphysical oscillations in the magnetic potential and field
components across shocks. Test computations are shown that confirm third order
accuracy for smooth test problems and high-resolution for test problems with
shock waves.Comment: 29 pages, 5 figures, 3 table
A Simple and Effective High-Order Shock-Capturing Limiter for Discontinuous Galerkin Methods
The discontinuous Galerkin (DG) finite element method when applied to
hyperbolic conservation laws requires the use of shock-capturing limiters in
order to suppress unphysical oscillations near large solution gradients. In
this work we develop a novel shock-capturing limiter that combines key ideas
from the limiter of Barth and Jespersen [AIAA-89-0366 (1989)] and the maximum
principle preserving (MPP) framework of Zhang and Shu [Proc. R. Soc. A, 467
(2011), pp. 2752--2776]. The limiting strategy is based on traversing the mesh
element-by-element in order to (1) find local upper and lower bounds on
user-defined variables by sampling these variables on neighboring elements, and
(2) to then enforce these local bounds by minimally damping the high-order
corrections. The main advantages of this limiting strategy is that it is simple
to implement, effective at shock capturing, and retains high-order accuracy of
the solution in smooth regimes. The resulting numerical scheme is applied to
several standard numerical tests in both one and two-dimensions and on both
Cartesian and unstructured grids. These tests are used as benchmarks to verify
and assess the accuracy and robustness of the method.Comment: 20 page
Positivity-preserving discontinuous Galerkin methods with Lax-Wendroff time discretizations
This work introduces a single-stage, single-step method for the compressible
Euler equations that is provably positivity-preserving and can be applied on
both Cartesian and unstructured meshes. This method is the first case of a
single-stage, single-step method that is simultaneously high-order,
positivity-preserving, and operates on unstructured meshes. Time-stepping is
accomplished via the Lax-Wendroff approach, which is also sometimes called the
Cauchy-Kovalevskaya procedure, where temporal derivatives in a Taylor series in
time are exchanged for spatial derivatives. The Lax-Wendroff discontinuous
Galerkin (LxW-DG) method developed in this work is formulated so that it looks
like a forward Euler update but with a high-order time-extrapolated flux. In
particular, the numerical flux used in this work is a linear combination of a
low-order positivity-preserving contribution and a high-order component that
can be damped to enforce positivity of the cell averages for the density and
pressure for each time step. In addition to this flux limiter, a moment limiter
is applied that forces positivity of the solution at finitely many quadrature
points within each cell. The combination of the flux limiter and the moment
limiter guarantees positivity of the cell averages from one time-step to the
next. Finally, a simple shock capturing limiter that uses the same basic
technology as the moment limiter is introduced in order to obtain
non-oscillatory results. The resulting scheme can be extended to arbitrary
order without increasing the size of the effective stencil. We present
numerical results in one and two space dimensions that demonstrate the
robustness of the proposed scheme.Comment: 28 pages, 9 figure
Finite Difference Weighted Essentially Non-Oscillatory Schemes with Constrained Transport for Ideal Magnetohydrodynamics
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, are treated as point values on the same mesh (i.e., there is no mesh
staggering); (2) both the spatial and temporal orders of accuracy are
fourth-order; (3) no spatial integration or multidimensional reconstructions
are needed in any step; and (4) special limiters in the magnetic vector
potential update are used to control unphysical oscillations in the magnetic
field. Several 2D and 3D numerical examples are presented to verify the order
of accuracy on smooth test problems and to show high-resolution on test
problems that involve shocks.Comment: 39 pages, 9 figures, 4 table
A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods
Nonlinear hyperbolic conservation laws admit singular solutions such as
shockwaves (discontinuities in conserved variables), rarefaction waves
(discontinuities in derivatives), and vacuum states (loss of strong
hyperbolicity). When ostensibly high-order numerical methods are applied in
such solution regimes, unphysical oscillations present themselves that can lead
to large errors and a breakdown of the numerical simulation. In this work we
develop a new Lax-Wendroff discontinuous Galerkin (LxW-DG) method with a
limiting strategy that keeps the solution non-oscillatory and
positivity-preserving for relevant variables, such as height in the shallow
water equations and density and pressure in the compressible Euler equations.
The proposed LxW-DG scheme updates the solution over each time-step with a
locally-implicit predictor followed by an explicit corrector. The
locally-implicit prediction phase is formulated in terms of primitive
variables, which greatly simplifies the solver. The resulting system of
nonlinear algebraic equations are approximately solved via a Picard iteration,
where the number of iterations is equal to the order of accuracy of the method.
The correction phase is an explicit evaluation formulated in terms of
conservative variables in order to guarantee numerical conservation. In order
to achieve full positivity-preservation, limiting is required in both the
prediction and correction steps. The resulting scheme is applied to several
standard test cases for the shallow water and compressible Euler equations. All
of the presented examples are written in a freely available open-source Python
code.Comment: 37 pages, 7 figures, 4 tables, 5 algorithm