用差分法与超松弛迭代法求高维平稳FPK方程的解

用不同精度的差分格式将高维平稳FPK方程离散化为线性代数方程组,然后用超松弛迭代法求解该线性代数方程组得到平稳FPK方程的近似解。讨论了不同的差分格式、网格密度及超松弛因子对解精度及收敛速度的影响,并与其他方法的计算精度进行比较,提出用多重网格算法提高计算效率。研究了典型的二维及四维随机系统的稳态响应,算例表明,该算法具有简洁、节省存储量且精度高的特点,是求解高维平稳FPK方程解的有效算法。

Stationary solutions of high dimensional reduced FPK equation using both finite difference method and successive over-relaxation method

The combination of finite difference method and successive over-relaxation method is employed to numerically get the stationary solution of high dimensional reduced Fokker-Planck-Kolmogorov(FPK)equation.The effects of different order of central difference scheme,the mesh density and factor of successive over-relaxation on the convergence and accuracy of numerical solution are discussed.A multi-mesh iterative method is proposed to improve the efficiency and accuracy of numerical solution.Two examples are given to illustrate the application of the proposed procedures.