High-performance implementation of a Runge–Kutta finite-difference scheme for the Higgs boson equation in the de Sitter spacetime

Abstract

High performance computations are presented for the Higgs boson equation in the de Sitter Spacetime using explicit fourth order Runge–Kutta scheme on the temporal discretization and fourth order finite difference discretization in space. In addition to the full, (3+1)−dimensional equation we also examine the (1+1)−dimensional radial solutions. The numerical code for the (3+1)−dimensional equation is programmed in CUDA Fortran and is performed on NVIDIA Tesla K40c GPU Accelerators. The radial form of the equation is 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 related semilinear Klein–Gordon equation with imaginary mass. Our numerical studies suggest several previously not known properties of the solution for the Higgs boson equation in the de Sitter spacetime 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.