带Ball-Chiu顶点的夸克Dyson-Schwinger方程的快速数值解
A Fast Numerical solution of the quark's Dyson-Schwinger equation with Ball-Chiu vertex
摘要：
本文提出了两种可行有效的方法来数值求解夸克的Dyson-Schwinger（qDSE），qDSE是由高自由度的第二类非线性积分方程组成的数学系统。由于qDSE的非线性和奇异性，很难解析求解。一般采用Gauss Legendre积分公式对奇异积分方程进行离散，然后用迭代法求出积分方程的近似解。这一过程的主要困难是未知函数，它是夸克在真空和有限化学势下的传播子，存在于积分符号的内部和外部。由于奇异性的存在，需要对积分符号内的未知函数进行高精度插值。通常传统的数值算例表明插值将花费大量的CPU时间。在这种情况下，我们提出了两种有效的方法来优化数值计算，一种是提出了一种改进的插值方法来代替传统的插值方法。此外，在现代科学计算中，广泛使用的另一种方法是在GCC中使用的OpenMP和自动并行化技术。最后，我们比较了不同算法的CPU时间，我们的数值结果表明所提出的方法的有效性。
英文摘要：
In this paper, we present two feasible and efficient methods to numerically solve the quark's Dyson-Schwinger (qDSE), the qDSE is mathematical systems of nonlinear integral equations of the second kind with high degrees of freedom. It is difficult to analytically solve the qDSE due to its non-linearity and the singularity. Normally we discrete the singular integral equation by Gauss Legendre integral integration formula, then the approximate solutions of integral equation are obtained by iterative method. The main difficulty in the progress is the unknown function, which is the quark's propagator at vacuum and at finite chemical potential, occurs inside and outside the integral sign. Because of the singularity, the unknown function inside the integral sign need to be interpolate with high precision. Normally traditional numerical examples show the interpolation will cost a lot of CPU time. In this case, we provide two effective and efficient methods to optimize the numerical calculation, one is we put forward a modified interpolation method to replace the traditional method. Besides, the technique of OpenMP and automatic parallelization in GCC is another method which has widely used in modern scientific computation. Finally, we compare CPU time with different algorithm and our numerical results show the efficiency of the proposed methods.
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