The Discontinuous Galerkin Method


Modeling of VLF waves in the magnetosphere still remains a computationally challenging task. Typical VLF waves may propagate for many seconds in the magnetosphere over scales much larger than a wavelength. In addition, nonlinear processes regularly give rise to observable phenomena such as triggering, entrainment, and growth of VLF waves. As such, computational modeling must be tailored to the problem at hand. On medium scales, layered media models are most appropriate. Over very large scales, the raytracing approximation is still the only tractable choice. For some types of problems, however, we must still resort to global solution methods.

The first computational method applied to propagation of radio waves in the earth's magnetosphere was arguably raytracing. A few decades ago, as computing power improved, so-called "moment methods" emerged as the method of choice for solving complicated radiation or scattering problems, but applying these methods to propagation in general media like plasmas still remained a significant challenge. Finite element methods (FEM) were also developed around this time and began to be used for problems with complicated material shapes, but development of explicit time-domain FEM for electromagnetics has progressed slowly. In more recent decades, finite difference time domain (FDTD) methods have become the method of choice, due to the ease with which inhomogeneities or complicated materials (like plasmas) can be handled. Interestingly, the FDTD method was first applied to electromagnetics by K.S. Yee as early as 1966 [Yee(1966)], but received little attention until RAM sizes became larger.

FDTD methods have been applied with great success to many complicated problems in electromagnetics, including plasmas, photonics, antennas, magnetospheric physics, etc. However, in recent years a more flexible family of methods called "discontinuous Galerkin finite element methods" (DG-FEM) have begun to gain acceptance in many areas that were traditionally the stronghold of FDTD.

Discontinuous Galerkin Method

In the finite difference method, the operators (the derivatives) are approximated:

$\displaystyle \frac{\partial u}{\partial x} \approx \frac{u(x_{i+1})-u(x_{i})}{\Delta x}$    

The DG method, on the other hand, approximates the solution, using a basis of expansion functions defined over a local finite element:

$\displaystyle u(x) \approx \sum_{i}^N u_i \phi_i(x)$    

In the nodal DG framework [Hesthaven and Warburton(2008)], the solution is solved at interpolation control points, as shown in the following figure:

Figure 1: DG finite element showing interpolation control points.
Image dgtriangle

Each element is connected to its neighbors via a quantity called a "flux," which is related to some physical or abstract conserved quantity in a system, e.g., conservation of mass in fluid mechanics, or conservation of power in electromagnetics.

Figure 2: DG finite elements. Element 2 is connected to its neighbors 1, 3, and 4 by exchanging flux at the faces between the elements.
Image connection

The entire domain is meshed to conform to the objects in the domain. For instance, in the following figure, we show a mesh conforming to a solid box with a small circle defined at the center:

Figure 3: A domain meshed around a circle defined at the center of a box.
Image mesh

The DG method has a number of interesting properties that make it extremely well-suited for solving complicated equations:

  • Conformal meshes: Since the solutions are expanded locally over each element, mesh elements can have any arbitrary shape dependent on the problem geometry.
  • Colocated fields: In contrast to FDTD, high accuracy does not require resorting to staggered grids.
  • Trivial parallelization: Parallelization only requires a way to communicate the fluxes at the faces between elements on different CPUs. That is, the stencil size is independent of the order of accuracy.
  • Arbitrary order of accuracy: The order of accuracy can be increased simply by increasing the number of basis functions in the expansion.

DG in plasmas

The DG method can easily be used in plasmas. A so-called ``cold plasma'' is characterized by a frequency-dependent plasma current:

$\displaystyle \begin{bmatrix}J_x\\ J_y\\ J_z\end{bmatrix} =  \begin{bmatrix} ...
...\sigma_{\vert\vert} \end{bmatrix} \begin{bmatrix}E_x\\ E_y\\ E_z\end{bmatrix}$    

In the nodal DG framework, all that is required is to convert this into a system of first-order ODEs, with plasma currents defined at the interpolation control points. A sample 2D simulation showing an antenna radiating into a magnetoplasma is shown below:

Figure 4: A simulation of an antenna radiating into a cold magnetoplasma. The magnetic field is oriented in the vertical direction. The antenna is oriented perpendicular to the plane of the plot.
Image highdetail


Real plasmas, of course, are not cold. Modeling all of the behavior of real plasmas is still a major computational challenge, since the entire velocity distribution of particles over the space must be taken into account (a 6-dimensional space!), and many nonlinearities can arise. Our current modeling efforts focus on using particle-in-cell (PIC) techniques within the DG framework to incorporate these effects. The PIC technique uses fields defined at fixed grid points to move particles freely around. In turn, these particles form a distribution of currents and charges, which are incorporated back into Maxwell's equations as extra source terms, as shown in the following figure:

Figure 5: Illustration of the classic PIC technique on a rectangular grid.
Image pic

The primary advantage of using PIC within the DG framework is that the approximate solution is defined everywhere, and is (in a least-squares sense) the closest to the actual solution we can get using a finite basis. We only need to reconstruct the solution using the basis in order to find the fields at any point. In contrast, in FDTD-PIC or finite volume-PIC, the fields must be interpolated, and this only weakly connects a value to the actual approximate solution.


Hesthaven and Warburton(2008)
Hesthaven, J. S., and T. Warburton (2008), Nodal Discontinuous Galerkin Methods.

Yee, K. (1966), Numerical solution of inital boundary value problems involving maxwell's equations in isotropic media, IEEE Transactions on Antennas and Propagation, 14, 302-307, doi: rm10.1109/TAP.1966.1138693.