Local averaged path integration method approach for nonlinear dynamic systems

Traditional path integration method always has a limited time step length, whose minimum is based on the subdivision degree of integral space. In order to numerically analyze a smaller time step probability evolution without significant increase in computing time, a new local averaged path integration (LAPI) method is investigated. The basic idea of this method is giving a local averaged value of transition probability density function (TPDF). Then the numerical integration of Chapman–Kolmogorov (CK) equation to the local averaged probability density function (PDF) will be calculated in lower space subdivision degree. Second-order partial derivative of PDF can be used to correct the deviation between local averaged PDF and the real one. It is demonstrated that the proposed method can give a more efficient calculation compared with traditional Gauss–Legendre path integration (GLPI). In the case of our method, smaller time step value is permitted with the same space subdivision degree, and smaller space subdivision degree is needed to keep the accuracy of stationary PDF with the same time step value. This merits also exist when analyzing transient PDF, which leads to a better capability of capturing the evolution of the PDF.