Fast Numerical Solution of a Kind of Nonlinear Integral Equations—Dyson-Schwinger Equations for Quark Propagator in Hadron Physics

Abstract

The nonlinear integral equation has been widely studied and has become the heart of the matter in many scientific and engineering fields, such as seismology, optical fiber evolution, radio astronomy, and hadron physics with Quantum Chromodynamics. The Dyson-Schwinger Equations (DSEs) approach provides an essential nonperturbative approach to investigating the properties of hadrons and hot/dense quark matter. Mathematically, the Dyson-Schwinger Equations are a group of coupled nonlinear integral equations of quark propagators, gluon propagators, ghost propagators, and various vertices. On account of the non-linearity and singularity of the coupled equations, it is almost impossible to solve the DSEs analytically. One has to resort to the numerical solution of the equations, in which efficient fast algorithms are key points in practice. In this work, two improvements for numerically solving the nonlinear and singular integral equation for quark propagator in a vacuum are proposed. One is a modified interpolation method for unknown functions in the integral with high degrees of freedom. The other is the parallelization on CPUs with OpenMP in GCC Comparing the CPU times with different algorithms, our results indicate that our proposed methods can greatly improve the efficiency and reduce the computation time of the CPU.