Introduction

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.

Introduction

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:

The DG method, on the other hand, approximates the *solution*,

using a basis of expansion functions defined over a local finite element:

In the nodal DG framework

[Hesthaven and

Warburton(2008)], the solution is solved at

interpolation control points, as shown in the following figure:

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.

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:

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:

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:

DG-PIC

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:

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.

Bibliography

- Hesthaven and Warburton(2008)

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

Galerkin Methods.- Yee(1966)

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.