Download Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes Daniel W. Swift

Advanced Methods for Space Simulations, edited by H. Usui and Y. Omura, pp. 77–89.
c TERRAPUB, Tokyo, 2007.
Generalized Curvilinear Coordinates in
Hybrid and Electromagnetic Codes
Daniel W. Swift
Geophysical Institute, University of Alaska, Fairbanks, Alaska 99775-7320, U.S.A.
This paper describes the elements for writing hybrid and electromagnetic plasma
simulation codes in generalized curvilinear coordinates. The coordinate system is
described by a table giving the three-dimensional Cartesian coordinate positions of
a structured curvilinear coordinate mesh. The paper shows how to calculate the geometric coefficients of the differential operators describing the electromagnetic field
and how to calculate the motion of particles across the curvilinear grid. In addition,
a new algorithm for calculating the particle current sources for the electromagnetic
field is described that conforms naturally to the curvilinear grid. Measures are described for checking and ensuring accuracy of the code. Extension of the methods
for multiple, discontinuously joined coordinate patches is also outlined.
1
Introduction
There are two main reasons for using curvilinear coordinates for space physics
applications. One is to conform to the particular geometry of the problem. A good
example is that of magnetosphere-ionosphere (MI) coupling. The Earth’s magnetosphere is in the shape of a comet, but one set of boundary conditions may be in the
ionosphere, which is a spherical shell. It might therefore be desirable to formulate
coordinate system with a spherical coordinate surface near the Earth, but which gradually deforms to roughly to a plane face of a box containing the simulation domain at
large distances. Figure 1 shows an example of such a coordinate system.
The other main reason is to accommodate disparate size scales. Again in the case
of MI coupling, one may want to resolve distances of tens of kilometers near the
Earth. Yet processes driving field-aligned currents near the Earth take place many
Earth Radii out in space, where it would be computationally expensive to maintain
that resolution. Also, one might want a relatively high degree of spatial resolution
across the plasma sheet and in the region of the bow shock and dayside magnetopause,
yet low resolution in the large volume occupied by the lobes of the magnetotail. An
example is shown if Fig. 2, which shows a global view in the midnight meridian plane
of the dawn-dusk component of the magnetic field in the top panel. The bottom panel
shows the field-aligned currents near the Earth.
Much of this presentation is based on an earlier publication (Swift, 1996) where
most of the techniques were introduced. The Swift (1996) paper, which focused on
the hybrid code, also describes how to include a cold ionospheric population in the
fluid approximation in the same spatial volume occupied by the discrete particles
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Fig. 1. An example of a coordinate system conforming to the surface of the Earth at the apex and deforming
to a plane boundary at large distances.
and how to subcycle the update of the fields to the particle push. This paper will,
however, focus on the curvilinear methods for doing field and particle operations
that are common to hybrid and electromagnetic codes and that can also be applied
to MHD codes. For the field operations, we will describe operations such as taking
the curl of a vector field, taking cross products and converting fields back and forth
between curvilinear and Cartesian coordinates. We shall also describe how to trace
the motion of a particle across a curvilinear grid and to calculate the current sources
of particles. The last section describes extension of curvilinear methods to multiple,
discontinuously joined coordinate patches.
2
Specification and Setup of the Grid
The coordinate grid is specified by a table giving the x, y, z-positions of the
coordinate points, qx (i, j, k, m), q y (i, j, k, m), qz (i, j, k, m), where the q’s specify
the x, y, z-positions of the curvilinear coordinate point denoted by the index i, j, k.
With this formulation, the grid can be changed simply by changing the defining table.
The last index, m takes of values of 1 or 2. The set corresponding to m = 1 defines
the location of the divergence of the magnetic field, B, and the other set of points
is the location of the divergence of the electric field, E. Also, the particle charge
density resides at the E-grid points. The grid contains no constraints to orthogonality,
because it has proven difficult to generate a grid that is both orthogonal and adapts
well to the geometrical constraints of the simulation.
Figure 3 shows a grid cell and the orientation and position of the electric and magnetic field components. This particular cell is termed a B-cell, since the divergence of
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
79
Fig. 2. Above shows the dawn-dusk component of the magnetic field in the midnight meridian plane. The
figure below shows a fine-scale view of the field-aligned currents near the Earth.
Fig. 3. A coordinate cell showing position and orientation of curvilinear components of the magnetic and
electric fields.
B is at the center of the cell. The components of B are on the cell faces. The E-grid
points are at the corners of the cell. If the center of the cell is located at the grid point
i, j, k, then the E-grid points are located at i −1/2 , j −1/2 , k −1/2 . In this picture, the
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Fig. 4. Schematic illustration of finite charge element carried by a particle on the grid.
components of E are located on the cell edges. In a view showing a cell with the
E-grid point at the center, the components of E would be on cell faces. The E- and
B-grid points are nested so differences are automatically centered.
3
Field Operations
Let us now look at how the field equations are set up. As a first example, consider
the update of the magnetic field, B from Faraday’s Law.
∂B
= −∇ × E.
∂t
(1)
Use Stokes’ Theorem to convert the curl operation to a line around a cell face.
∂B
= − E · dl.
∂t
(2)
In curvilinear coordinates, B is represented by
B = B 1 e1 + B 2 e 2 + B 3 e3
(3)
where the ei ’s are unit vectors parallel to cell edges. Making use of (2) and (3), the
curl operation can be evaluated. The result applied to the right-hand face of Fig. 1 is
n+1
B1
1
e1 · A = En+1 · l2 i+ 1 , j,k− 1 − En+1 · l2 i+ 1 , j,k+ 1
2
2
2
2
t
n+1 n+1 + E
· l3 i+ 1 , j+ 1 ,k − E
· l3 i+ 1 , j− 1 ,k
2
2
2
2
where
(l2 )i+ 12 , j,k+ 12 = ri+ 12 , j+ 12 ,k+ 12 − ri+ 12 , j− 12 ,k+ 12
(4)
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
81
where the grid point indices are specified by i, j, k and n is the time level. The
x-component of l2 can be computed from the table specifying the location of the
coordinate points and is given by
l2x = qx (i, j + 1, k, 2) − qx (i, j, k, 2) .
(5)
The unit basis vectors, and the cell-face area, A1 are given in terms of the tangent
basis vectors li , i = 1, 2, 3 are given by
e1 = l1 /l1 ,
A 1 = l 2 × l3 .
(6)
Using this formulation of the curl operation yields a vector that is exactly divergenceless. This can be seen from Fig. 1. The divergence at the center of the cell can be
shown to be the sum of the line integrals around all the faces of the cell. The line
integrals along cell edges on adjacent faces cancel each other.
In a non-orthogonal coordinate system, there are two sets of basis vectors. One is
the tangent basis vector, which is given above. The other basis set is normal to cell
faces. Thus the vector in (3) can also be represented
B = e1 B1 + e2 B2 + e3 B3 .
(7)
Where
l 2 × l3
j
ei · e j = δi .
(8)
l1 ,
(l1 · (l2 × l3 ))
Those who have some familiarity with tensor analysis will recognize the raised indices are related to contravariant components, while the lowered indices are covariant
components. The difference is that we choose to use dimensionless unit basis vectors. It can be seen that the components of B with raised indices are cell-face normal
components, while those with lowered indices are cell-edge tangent components. The
component indices can be raised by taking the dot-product of the expression in (7)
with the cell-face normal basis vectors in (8) and lowered by taking the dot-product
of the expression in (3) with the cell-edge basis vectors in (6).
The equations involved in the hybrid code require the taking of cross products of
vectors with cell-face (raised index) components. For example,
u×B = [e1 · (e2 × e3 )] [e1 u 2 B 3 − u 3 B 2 +e2 u 3 B 1 − u 1 B 3 +e3 u 1 B 2 − u 2 B 1 ].
(9)
Notice, this cross product yields a vector with cell-edge components, as required for
the curl operation.
We also need to convert back and forth between Cartesian and curvilinear components. From the relation
e1 =
B = 1x Bx + 1 y B y + 1z Bz = e1 B 1 + e2 B 2 + e3 B 3 .
(10)
The components are easily derived, for example,
Bx = 1x · e1 B 1 + 1x · e2 B 2 + 1x · e3 B 3
B 1 = e1 · 1 x B x + 1 y B y + 1 z B z .
(11)
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The above has provided a sampling of the types of operations that go into development of a hybrid code. All the coefficients are derived from the tangent vectors
given in (4) and (5).
4
Particle Motion
The particle phase space coordinates are the position specified in curvilinear coordinates and the velocity specified in Cartesian components.
P = q 1 , q 2 , q 3 , vx , v y , vz .
(12)
Where the q’s are the curvilinear position of a particle, and the v’s are the Cartesian
components of the velocity. Topologically, the curvilinear grid is a cubic lattice with
unit distance between grid points. Thus, the index of the cell containing the particle is
obtained by simply truncating the particle index. For example in FORTRAN notation
I = INT(Q1) + 1, and similarly for the other indices. The velocity is represented in
Cartesian coordinates in order to avoid explicit calculation of derivatives of the metric
coefficients given (4). Those familiar with tensor analysis would recognize that those
derivatives would comprise elements of the three-index Christoffel coefficients.
The relation between the rate of change of the curvilinear position and the particle
velocity can be derived from
v = 1x vx + 1 y v y + 1z vz = l1 q̇ 1 + l2 q̇ 2 + l3 q̇ 3 .
(13)
From which the components of curvilinear velocity are readily obtained
q̇ i =
ei · v x 1 x + v y 1 y + vz 1z .
li
(14)
The Cartesian components of ei are obtained (8) making use of (4) and (5).
5
Field Sources
The customary method of calculating field sources from particle positions and
velocities is the particle-in-cell weighting in which a fraction of the particle charge
(PIC) is assigned to each of the eight grid points on the corners of the cell. The field
sources in both hybrid and electromagnetic codes are the electric currents derived
from particle fluxes. This procedure yields Cartesian components of the flux at the
cell corners, which has to be converted to cell-face normal curvilinear components.
In addition, the current components have to be interpolated from cell corners to cell
faces. Experience indicates this procedure works satisfactorily in a hybrid code; but
this procedure is not very satisfactory in electromagnetic codes. The reason is that
charge is not well conserved. This inaccuracy can result in an intolerable level of
noise. In strictly Cartesian coordinates with regular boundaries, one can do a divergence clean on the electric field by inversion of the Laplace operator. This is very
difficult in Curvilinear coordinates, where the Laplace operator has variable coefficients.
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
83
Villasenor and Buneman (1992) have proposed an algorithm that conserves
charge exactly and that conforms naturally to a curvilinear coordinate grid and places
the current on cell faces where it is needed in the field equations.
The scheme is illustrated schematically in Fig. 2. As mentioned previously, the
grid is topologically equivalent to a square or cubic lattice. Each particle carries with
it a square or cubic charge element, and the particle transports this charge as it moves
through the grid. The current is simply the amount of charge that moves across a
cell face in a time step divided by the time step The current comes out in cell-face
components exactly as needed for updating the field in curvilinear coordinates.
The algorithm as described in the original Villasenor-Buneman (1992) paper describes a very complicated ten-branch logic tree. In the electromagnetic code, the
time step is constrained by the Courant condition with respect to the velocity of light,
and a particle is constrained by relativity from moving faster than the speed of light.
Therefore, the number of cell boundaries a particle charge element can cross is constrained in the electromagnetic code. There is no such rigid constraint in the hybrid
code. The hybrid code we have developed uses a modified and conceptually much
simpler variation of the Villasenor-Buneman algorithm. In the subroutine where the
particles are moved, the time step for each is broken into variable increments which
end when a particle charge element encounters or leaves a new cell boundary. The
current contributions in each of these increments is summed. In most instances, the
particle will cross fewer than ten cell boundaries, so this represents a saving over
the original algorithm, but the algorithm also has the flexibility to accommodate the
occasional fast particle that cuts across cell corners.
6
Tests and Other Practical Considerations
The most common field boundary condition used is the specification of two tangential components of the electric field and the normal component of the magnetic
field. These conditions require that the E-grid points lie on the outer boundary. Thus
the generation of the grid would begin with specification of the E-grid points. The
B-grid point would then be placed in the center of cells formed by the eight E-grid
points. Thus the advancement of B from Faraday’s law would have second order
spatial accuracy. The problem is that in a variable grid, the E-grid points are not
necessarily in the center of B-grid cells, so the computation of curl B would not be
second-order accurate. This can be corrected by a relaxation process of centering the
E-grid points in the B-grid. Very close to second-order accuracy can be achieved by
about two cycles of grid relaxation, while keeping the E-grid boundary points fixed.
It can be seen that the accuracy is dependent on the smoothness of the grid, so it is
wise to avoid sudden changes in orientation of the grid lines or changes in grid density. If the original has large changes, multiple relaxation cycles will serve to smooth
the grid.
Although the methods developed have accommodated non-orthogonal grids, experience has shown that the code performs poorly if the angle between tangent vectors
becomes less than 45◦ .
There are several accuracy checks on the various modules that should be per-
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D. W. Swift
Fig. 5. A meridian-plane cut showing the use of multiple coordinate patches for a global-scale simulation
of the magnetosphere.
formed in the process of developing a curvilinear code. The most critical test consists
of taking the a vector like A = 1x y. Convert it to a cell-face normal curvilinear representation. Then convert the result to a cell-edge tangent representation and take the
curl. Next convert the result back to Cartesian. The result should be the unit vector
1z . Experience indicates satisfactory code performance with a cumulative accuracy
of 1%.
Another check is to trace the force-free motion of a particle across the grid. That
should be a straight line. One can use the coordinate tables and PIC weighting to
compute the instantaneous particle position in Cartesian coordinates that can be compared to an analytically calculated position.
The Villasenor-Buneman conservative algorithm should be checked by tracing
the density of a single moving particle. This is done by taking the divergence of the
particle flux and integrating with respect to time. If the density is initialized to the
density calculated by PIC weighting at t = 0, the result is the density calculated by
PIC weighting of the particle charge at the final position of the particle.
Finally, those who have worked with hybrid codes know they can be quite temperamental. Hybrid codes can become unstable or otherwise produce inaccurate results
if values are not properly centered for second-order accuracy. One aspect that has
been glossed over is that of interpolation that is necessary between parts of a cell in
converting between cell-face and cell-edge components of a vector, and in converting
between curvilinear and Cartesian representations. Care must be taken to maintain
second-order accuracy. Second-order accuracy also requires calculation of dual sets
of tangent basis vectors formed by separately differencing between E- and B-cell
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
85
Fig. 6. The coordinate grid on the ionospheric shell of the Earth. The center is occupied by the North
polar patch, which is surrounded by the day, night, dawn and dusk patches. Notice the small overlap
between patches.
grid points so that the vectors are centered on the line between grid points. Cell-face
areas must also be carefully centered on the cell face.
7
Extension to Multiple Coordinate Patches
Figure 1 shows a coordinate patch suitable for simulating a segment of the magnetosphere that extends from the ionospheric outward to fifteen Earth radii. A truly
global simulation would require the joining of several coordinate patches to cover
the region surrounding the Earth and to include the magnetotail. Figure 5 shows how
several of the coordinate patches shown in Fig. 1 may be joined for a global scale simulation. This figure shows a meridian plane cut of the dayside, nightside, north and
south polar and tail patches. Not shown are the dawn and dusk coordinate patches.
Six of the patches bound on the ionospheric shell.
The multiple patch simulation is considerably more complicated than one based
on a single curvilinear coordinate system. The major complication arises from the
fact that the coordinate system undergoes discontinuous changes from one coordinate
patch to another. This means that components of the field change in crossing a patch
boundary. The reader will also note there is a greater density of grid points on the
dayside patch than on the neighboring patches. This provision is added to capture
details of the bow shock and magnetopause.
Figure 5 shows the B-grid coordinate lines The outer B-grid coordinate surfaces
must coincide with the outer B-grid coordinate surfaces of the neighboring patches.
As mentioned in the previous section, boundary conditions are imposed on the surfaces defined by the outer layer of E-grid points. On the interface between patches,
these points lie inside the domain of a neighboring patch. This overlap is illustrated
in Fig. 6, which shows the E-coordinate grid on the ionospheric surface. Notice the
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D. W. Swift
small overlap between patches. Although the grid lines normal to the patch boundary
do not necessarily coincide, the density of grid surfaces is constrained to be constant
across patch boundaries, so that the outer E-grid surface nearly coincides with next to
outermost surface on the adjacent patch. Boundary conditions are imposed by interpolating field data from points on the outer B-grid and the next-to-last E-grid points,
called source points, to the outer E-grid points of the neighboring patch, which will
be called target points.
To impose inter-patch boundary conditions, it is first necessary to identify the
closest source points surrounding a given target point. The strategy for doing this is
to a large extent dependent on the details of the coordinate system and will not be described here. However, as mentioned above, the vector fields must be interpolated in
both position and direction. To do this, we realize that a vector field can be expanded
in Cartesian coordinates about a target point (x0 , y0 , z 0 )
E = 1x [c1 (x − x0 ) + c2 (y − y0 ) + c3 (z − z 0 ) + c4 ]
+ 1 y [c5 (x − x0 ) + c6 (y − y0 ) + c7 (z − z 0 ) + c8 ]
+ 1z [c9 (x − x0 ) + c10 (y − y0 ) + c11 (z − z 0 ) + c12 ] .
(15)
The coefficients, ci may systematically be determined by computing the curvilinear
components at twelve source points surrounding the target point. For example.
E 1 = e1 · E
(16)
where e1 is given by (8). Upon taking the scalar product of E in (15) with e1 at the
source point (xi , yi , z i ), we get the relationship
E 1 = T1x [c1 (xi − x0 ) + c2 (yi − y0 ) + c3 (z i − z 0 ) + c4 ]
+ T1y [c5 (xi − x0 ) + c6 (yi − y0 ) + c7 (z i − z 0 ) + c8 ]
+ T1z [c9 (xi − x0 ) + c10 (yi − y0 ) + c11 (z i − z 0 ) + c12 ] .
Where, for example, T1x = e1 · 1x . This process is repeated for all three components
(E 1 , E 2 , E 3 ) using four source points for each component. This generates twelve
equations for the twelve coefficients c j . Care has to be taken that the source points
form a non-degenerate tetrahedron encompassing the source point. The coefficients
ci can then be found by matrix inversion. The target field component can then be
determined by application of an operation like in (16) to the expression in (15). The
coefficients ci depend only on the geometry, of the coordinate system which does not
change with changing values of the source field. Therefore, it is possible to set up a
simple linear relationship between the source point fields and target point fields, E ti
E ti =
H ji f is .
j=1
Where f s is one of the E k source points, and the H ’s are precomputed.
(17)
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
87
There are instances in which the boundary conditions require edge-aligned components. In that case, one should take the scalar product of the expression in (15)
with ei , as given in (6). One can similarly apply the expression in (15) to compute the
coefficients ci if the source fields are known in terms of edge-aligned components.
There is an additional complication when field target points are on the edge of a
coordinate patch that is not part of an external boundary. In that case, source data must
be taken from two coordinate patches simultaneously. However, the above-described
interpolation procedures still apply.
When a particle crosses a coordinate patch boundary, it’s curvilinear coordinate
position must be changed in a way that its actual spatial position remains unchanged.
This requires a table lookup and interpolation in a manner not dissimilar to the procedure for identifying field source points with target points. The velocity components,
being Cartesian, do not change. However, use of the Villasenor-Bunemann current
algorithm requires care in order to maintain charge conservation.
As mentioned above, there is an overlap between coordinate patches. Say, a particle is passing from coordinate Patch 1 to Patch 2. A particle enters the domain of
Patch 2 before it leaves Patch 1. At this point the particle is placed in Patch 2, so
it resides simultaneously in both patches and is tracked in both patches. It is finally
eliminated from Patch 1 when it passes through the outer E-grid surface of Patch 1.
There will also be instances when a particle passes close to a coordinate grid edge. In
this case, the procedure allows particles to exist in three patches simultaneously.
A global-scale hybrid code is of such a size that it requires use of a massively
parallel computer. It is therefore logical that particle and fields residing in different
coordinate patches be processed by separate processors. The coordinate system outlined in Fig. 5 consists of seven coordinate patches. It is desirable to use a much
larger number of processors. The number of particles in each of these patches is significantly different. Since most computer time is devoted to pushing of particles, it
is desirable to have the flexibility to independently partition the various coordinate
patches into a varying number of segments to achieve particle balance among processors without. It should also be possible to independently repartition the coordinate
patches without having to do any reprogramming.
This is accomplished by doing the message passing is done in two stages. First,
data is passed among segments interior to coordinate patches. This is easily done
since within a patch there is a common coordinate system topologically equivalent to
a cubic lattice. Next, each segment that resides on a coordinate patch boundary passes
its field data to a buffer which handles data from all segments on the coordinate patch
face. The field interpolation procedures described above for an entire coordinate
patch face are carried out, and the data is passed to a buffer on a processor in the
neighboring coordinate patch. Then processors for each of the segments on the patch
face claim its own section of data. Similarly, particles crossing a patch boundary are
put in a common buffer for an entire patch face and then passed to a buffer on for
the adjoining patch. The particles are then distributed to segments according to their
position.
The testing procedure is much as described in Section 6. The curl of a vector field
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D. W. Swift
like A = 1x y should yield a constant unit vector everywhere, including points on
and near patch boundaries. The force-free trajectory of a particle should be a straight
line, even when crossing coordinate patch boundaries, and the total charge carried by
a particle should remain the same as it crosses coordinate patch boundaries.
Finally, a word about diagnostics: Each processor writes out files containing field
data within its spatial domain. Standard routines are available for converting the
fields to Cartesian components. Commerical visualization packages contain modules
for plotting volumetric data from unstructured grids. Many of these procedures for
unstructured grids are impossibly slow for any grid size approaching 100 × 100 × 100
grid points. Then there is the problem of collating data and preparing displays for
regions that span volumes occupied by many processors.
There are much faster procedures that take advantage of the structured nature
of the curvilinear grid and integrate data from all processor domains within a predetermined volume. The number of desired Cartesian grid points and the resolution
of the output grid can also be specified. A processing program can be set up to
loop through the files written by the various processors. The program then loops
through each of the curvilinear grid points represented in a processor file. For each
curvilinear hexahedral cell, the program determines the maximum and minimum x,
y and z-values of the box containing the curvilinear cell and then determines the
indices of the Cartesian grid points within that rectangular box. The program then
loops trough the indices of Cartesian points within the box and finds the curvilinear
points closest to each Cartesian point. Next, it determines whether the Cartesian point
is within the curvilinear cell. If it is, it then does a four-point linear interpolation
from the closest curvilinear point and three neighboring corner points on the cell.
An alternative version does an eight-point tri-linear interpolation utilizing all corner
points defining the curvilinear cell. This procedure scales linearly with the number
of points. Moreover, it has the flexibility to allow the user to render data in any
predetermined box at any resolution. Thus t is possible to see details in the data near
the ionosphere in one view and to look at larger scale structure in more distant parts
of the magnetosphere.
8
Concluding Remarks
Curvilinear coordinates are obviously more complicated than a Cartesian grid,
and the computational resources needed to update the fields and push the particles
is certainly much greater. Curvilinear coordinates have an advantage in large-scale
problems where high spatial resolution is needed in only some regions. Curvilinear
coordinates avoids the necessity of carrying a high density of grid points were the
spatial resolution is not needed.
Since the grid is represented by a table, the grid can be changed by changing
the defining table without the necessity of reprogramming. This opens the way for
adaptive meshes. However, a change in the underlying grid in the course of a simulation will require the repositioning of the particles and interpolation, and possibly
re-orientation of the field components, to the old to new grid.
The other and more flexible alternative to the curvilinear grid is the unstructured
Generalized Curvilinear Coordinates in Hybrid and Electromagnetic Codes
89
mesh. Curvilinear coordinates have the connectivity of a cubic lattice, so that that
every cell knows its nearest neighbors and every particle knows which grid cell it is
in. In situations of where there are many sharp boundaries, the unstructured mesh may
be a necessity. However, the conceptual simplicity of the curvilinear coordinate offers
a clear advantage in space physics problems where boundaries may be represented by
a small number of smooth surfaces.
Curvilinear coordinate techniques are now being extended to systems composed
of multiple, discontinuously joined, coordinate patches. The multiple coordinate
patches, among other things, makes it possible to cover the entire surface of the Earth
in the coordinate grid without singularities. The multiple-patch technique is being
used in a global-scale hybrid code model of the Earth’s magnetosphere.
Acknowledgments. This work was supported by the USA National Science Foundation under
Grant 0087854. Computer resources are provided by the Arctic Region Supercomputer at the
University of Alaska.
References
Swift, D. W., Use of a hybrid code for global-scale plasma simulation, J. Comput. Phys., 126, 109–121,
1996.
Villasenor, J. and O. Buneman, Rigorous charge conservation for local electromagnetic field solvers, Comput. Phys. Commun., 2, 306–316, 1992.