  High performance computations are presented for the Higgs Boson Equation in
the de Sitter Space- time using explicit fourth order Runge-Kutta scheme on the
temporal discretization and fourth order finite difference discretization in
space. In addition to the fully three space dimensional equation its one space
dimensional radial solutions are also examined. The numerical code for the
three space di- mensional equation has been programmed in CUDA Fortran and was
performed on NVIDIA Tesla K40c GPU Accelerator. The radial form of the equation
was simulated in MATLAB. The numerical results demonstrate the existing
theoretical result that under certain conditions bubbles form in the scalar
field. We also demonstrate the known blow-up phenomena for the solutions of the
semilinear Klein-Gordon equation with imaginary mass. Our numerical studies
suggest several previously not known properties of the solution for which
theoretical proofs do not exist yet: 1. smooth solution exists for all time if
the initial conditions are compactly supported and smooth; 2. under some
conditions no bubbles form; 3. solutions converge to step functions related to
unforced, damped Duffing equations.
