A modified Chebyshev collocation method for the generalized probability density evolution equation: Fid-Bau Portal

A modified Chebyshev collocation method for the generalized probability density evolution equation

Tian, Rui / Xu, Yazhou

Abstract Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

Highlights • A modified Chebyshev collocation method has been proposed for solving the generalized density evolution equation. • The modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency. • The modified Chebyshev collocation schemes could effectively suppress numerical dispersion and numerical dissipation.