To the left, the first four B-splines , which define the particles' weight functions. These four are all accessible in the code. To the right, the cell (area) weighting of the interpolation kernel for the triangular-shaped-cloud. This shape function is the piecewise linear spline, whereas the piecewise quadratic spline shape produces the bell-shaped (cubic spline) area weighting function—cf. Eq. (14) .
Spatial staggering in the PHOTON-PLASMA code.
Time staggering and integration order in the PHOTON-PLASMA code. and J are staggered backwards in time, , while everything else is time centered.
Example: 6th order difference operation in 1D (along the y-axis). Due to the Yee mesh staggered layout of variables, the central difference is computed exactly where needed. The differential operator , called “ddyup” (“ddydn”) in the code, produces the 6th order spatial derivative w.r.t. the y-axis, up-shifted (down-shifted) one half mesh point on the y-axis. For the example in the figure, this produces the correct time derivative of Bx at the desired mesh point location, . In the nomenclature adopted here, this operator is denoted . Cell centers are marked in blue, and cell edges in red. Further, compare this figure with the respective components in Fig. 2 .
Sketch of the scattering mechanism in the code of an incident macro particle (dark blue) on a target macro particle (green) resulting in the creation of a scattered fraction of the original macro particle pair, and a pair of unaffected macroparticles carrying the remaining fraction of the original particle pair. The process is shown both for laboratory (or simulation) rest frame (top panel), and for the target rest frame (bottom panel).
Changes in electron (upper panels) and ion (lower panels) temperatures in the numerical heating experiment as a function of time for different choices of integration scheme ordering. Left (right) panels show temperature increase (decrease), for 5 (50) particles per species per cell. A heating rate, H(t), is defined in proportion to the slope of the temperature dependence, T(t), or to the square root of the total energy, . Hence the heating rate becomes, for example, . Time is given in units of , the inverse electron plasma frequency.
Top panels: energy conservation and stream velocity in the original rest frame. Bottom panels: temperature evolution in the parallel (left) and perpendicular (right) beam direction—, i.e., and . For reference, results are also included using a simple FDTD explicit solver for the electromagnetic fields, and using a much higher resolution. Notice that the runs have been done without applying any kind of filtering to the current density.
Fourier power in the out-of-plane magnetic field at . To extend the dynamic range, the power is shown with the fourth root: . The panels from left to right are TSC and cubic interpolation with 2nd order field solver, cubic interpolation with 6th order field solver, and CCo method with 6th order field solver and 2nd or 6th order time integration.
Growth of the volume-totaled electrostatic energy, for the electric field projected onto the direction of propagation of the fastest growing mode, . Runs 1-6 are compared in the plot. Thickened line segment run 1 (black) gives the fitting interval. Runs have decreasing initial energy for increasing run number designation (Table II ). Runs 4 (orange diamonds), 5 (yellow diamonds), and 6 (green triangles) are almost completely coinciding.
Left panel: Magnetic field density normalized to the upstream kinetic energy density at , shown in a 500 cutout near the shock interface. To enhance the dynamic range, a signed is shown. Right panel: Corresponding phase space density. Notice how not only the phase space is quenched by the cooling but also the magnetic field density at the shock interface, and downstream of the shock it depends on the cooling rate. The velocity range is different for the different cases.
Top panel: Density ratio between up and down stream at , as a function of cooling time. Bottom panel: Effective adiabatic index, derived from the shock velocity. The rightmost point is for a run without radiative cooling.
The particle distribution function sampled in the downstream region for the case of no cooling and , firmly in the weakly radiative regime. The power law index is indicated with the dashed line, and given in the legend.
Early non-linear (left) and late non-linear vortex paring (right panel) stages of a PIC code model of the Kelvin-Helmholtz instability. The small scale electron density waves barely visible in the left panel are probably because of the initial condition not being a perfect kinetic equilibrium. 48,49
Early non-linear (left) and late non-linear vortex paring (right panel) stages of a MHD code model of the Kelvin-Helmholtz instability.
Density profile of a 2D collisionless shock. The dashed lines indicate the z-boundaries of MPI domains. They are updated dynamically to equalize the load.
Weak scaling in pure MPI mode when running on the BG/P machine JUGENE. The setup is a relativistic two-stream experiment with periodic boundaries and a streaming motion. There are roughly 80 particles per cell with 163 cells per MPI domain. This experiment was not using the charge conserving current deposition, and the scaling of the current code is significantly better.
Weak scaling in hybrid mode when running on a 16 core E5-2670 2.6 GHz Sandy Bridge node and the BG/Q machine JUQUEEN. The setup is a relativistic two-stream experiment with periodic boundaries and a streaming motion. There are 60 particles per cell. This experiment is using the charge conserving current deposition. The label gives number of OpenMP threads times MPI threads, and on both machines, we take advantage of hyperthreading, while performance is measured per physical core. The left panel shows weak scaling for a fixed domain size of 163 cells per thread. The middle panel shows weak scaling on 64 nodes or 4096 threads on JUQUEEN with only 123 cells per thread. The right panel shows strong scaling keeping the domain constant at 503 cells.
Resolution study for the cold beam instability.
Schemes order variation in the relativistic mixed-mode two-stream instability test case, for finite difference operators (fields/sources), shape function (particle-mesh/mesh-particle interpolation), charge-conservation, time integration order.
Growth rates measured for the six runs listed in Table II for the two cases of the MMI and TSI.
Growth rate, for the magnetized KHI; comparison between identical runs with the PHOTON-PLASMA code and the Stagger code.
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