奇异摄动反应扩散方程数值模拟的粒子群优化算法

针对Shishkin网格方法在数值求解奇异摄动反应扩散方程时，网格过度点参数的选取具有不确定性的缺陷，提出了一种用粒子群优化(PSO)算法估计Shishkin网格参数的方法。首先基于有限差分方法，构造了以误差范数最小为目标的无约束优化问题，并用PSO算法进行了求解。该方法克服了人为选择参数的缺陷。实验结果表明：与单纯形算法相比，PSO算法在优化Shishkin网格参数时能够收敛到全局最优解;而且在最优网格参数下，奇异摄动反应扩散方程的数值结果在边界层的精度也得到了明显提高，进一步说明了所提方法的有效性和可行性。

Numerical simulation-based particle swarm optimization algorithm of singularly perturbed reaction-diffusion equation

There are some uncertainty defects of parameter selection in excessive grid points, when Shishkin mesh method is used to solve singularly perturbed reaction-diffusion equations by numerical solution. The Particle Swarm Optimization (PSO) algorithm was used to estimate the parameters of Shishkin mesh. First, based on finite difference method, an unconstrained optimization problem which was directed towards minimizing error norm was constructed. Then the PSO was used to solve it. The proposed method overcame the defects of artificial parameters selection. The experimental results show that compared with the simplex algorithm, PSO can converge to the global optimal solution for optimizing the parameters of Shishkin mesh. The accuracy of numerical computing is improved significantly for singularly perturbed reaction-diffusion equations on the boundary layer, which further illustrates the effectiveness and feasibility of the proposed method.