Abstract

CHAPTER 3. BIFURCATION ANALYSIS 58
To handle the nonlinearity in computing W ( f), each processor computed the elec-
tron density n(x) (1.11) for each x-point it owned. The processors would then send
theirpartofn(x) to one processor, which is designated the main processor. This
main processor takes all of n(x), performs the Poisson solve to compute the potential
energy U(x) (1.13), and sends out U(x) to all the processors. Once each processor
had U(x), it could compute P ( f) for each of its x-points. To compute the derivative
term in K( f), each processor would need to know the f values on the 2 x grid points
before its smallest x-point and the 2 x grid point ahead of its largest x-point since a
second-order upwind differencing scheme was used. These values were passed between
the processors. Finally, each processor would add these terms up to get the W ( f)
evaluation on the parts of the domain the processor owned.
To demonstrate the parallel efficiency of our program, the simulation with Nx =
512 and Nk = 2048 was run using from 2 up to 80 processors. The runs reported
in this section were performed on processors of a Linux cluster at Sandia National
Laboratories. This cluster has a total of 236 compute nodes. The nodes are dual
3.06 GHz Xeon processors, each with 2 GB of RAM. The table below compares the
run times for taking 5 continuation steps, from V =0.2093 to V =0.2293. Since
the nodes used to perform the efficiency study are dual processor, we decided a fair
evaluation of the efficiency required a base case of 2 processors instead of the normal
1 processor. The communication between 2 processors on the same node would be
more efficient than the communication between processors across distinct nodes.