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9.3.5 One-Dimensional Electromagnetic Code

The parallel one-dimensional electrostatic code was modified to include the effects of external and self-consistent magnetic fields. This one-dimensional electromagnetic code, with kinetic electrons and ions, has been used to study electron dynamics in oblique collisionless shock waves such as in the earth's bow shock. Forces on the particles are found from the fields at the grid points by interpolation. For this code, with variation in the x direction only, the orbit equations for the iparticle are

Motion is followed in the x direction only, but all three velocity components must be calculated in order to calculate the force. The longitudinal (along x) electric field is found by solving Poisson's equation

The transverse (to x) electromagnetic fields, , , , and , are found by solving

Table 9.2: Comparison of Push Times on Various Computers

The plasma current density and charge density are found at the grid points by interpolation from the particle positions. Only the transverse (y and z) components of the plasma current are needed. These coupled particle and field equations are solved in time as an initial value problem. As in the electrostatic code, the fields are solved by Fourier-transforming  the charge and current densities and solving the equation in k space, and advancing the Fourier components in time. External fields and currents can also be included. At each time step, the fields are transformed back to configuration space to calculate the forces needed to advance the particles to the next time step. The hypercube FFT  routine described in Section 12.4 was used in the one-dimensional codes. Extending the existing parallel electrostatic code to include the electromagnetic effects required no change in the parallel decomposition of the code.

Guy Robinson
Wed Mar 1 10:19:35 EST 1995