  Particle tracking simulations with space charge effects are very important
for high-intensity proton rings. Since they include not only Hamilton mechanics
of a single particle but constructing charge densities and solving Poisson
equations to obtain the electromagnetic field due to the space charge, they are
extremely time-consuming. We have newly developed a particle tracking
simulation code that can be used in Graphics Processing Units (GPU). GPUs have
strong capacities of parallel processing so that the calculation of
single-particle mechanics can be done very fast by complete parallelization.
Our new code also includes the space charge effect. It must construct charge
densities, which cannot be completely parallelized. For the charge density
construction, we can use shared memory which can be accessed very fast from
each thread. The usage of shared memory is another advantage of GPU computing.
As a result of our new development, we increase the speed of our particle
tracking including space charge effect approximately 10 times faster than that
in the case of our conventional code used in CPU.
