Adaptive Discontinuous Galerkin Method for Response-Excitation PDF Equations

Evolution equations of the joint response-excitation probability density function (REPDF) generalize the existing PDF evolution equations and enable us to compute the PDF of the solution of stochastic systems driven by colored random noise. This paper aims at developing an efficient numerical method for this evolution equation of REPDF by considering the response and excitation spaces separately. For the response space, a nonconforming adaptive discontinuous Galerkin method is used to resolve both local and discontinuous dynamics while a probabilistic collocation method is used for the excitation space. We propose two fundamentally different adaptive schemes for the response space using either the local variance combined with the boundary flux difference or using particle trajectories. The effectiveness of the proposed new algorithm is demonstrated in two prototype applications dealing with randomly forced nonlinear oscillators. We first study the stochastic pendulum problem and compare the resulting PDF against the one obtained from Monte Carlo simulation. Subsequently, we study the Duffing oscillator for two different types of stochastic forcing and random initial conditions. We observe both oscillatory and chaotic dynamics and compare the results against the solution of the effective Fokker–Planck equation. The framework we develop here is general and can be readily extended to stochastic PDEs subject to random boundary conditions, random initial conditions, or random forcing terms.