Adaptive Numerical Solutions of Fokker-Planck Equations in Computational Uncertainty Quantification

摘要

In this work the computational uncertainty quantification problem is formulated in terms of diffusionless Fokker-Planck equations and adaptive finite difference solutions are applied to obtain the time evolution of response probability density functions (PDPs). Two adaptive algorithms are developed for obtaining computational meshes which conform to the time evolving probability distribution. The results for the moments of the response variables are obtained through numerical integration of the obtained probability distributions. Results from each of these methods are compared to a fixed domain, non-adaptive finite difference solution, analytical solutions (when they exist) and Monte Carlo simulation to asses the impact, both in terms of accuracy and efficiency, of applying the adaptive schemes. These comparisons indicate that 1) formulating the computational uncertainty quantification problem in terms of Fokker-Planck equations yields accurate solutions for problems which are time dependent and nonlinear and 2) the adaptive methods give accurate solutions with an a order of magnitude decrease in the number of grid points as compared to the fixed domain solutions. These promising results indicate the possibility for applying the adaptive methods to overcome some of the open issues in computational uncertainty quantification including long time integration and discontinuous (in random space) response variables.