New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new types of differential equations satisfied by the joint response excitation probability density function associated with the stochastic solution to first-order nonlinear scalar PDEs. The theory is developed for arbitrary fully nonlinear and for quasilinear first-order stochastic PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are presented for a nonlinear advection problem with an additional quadratic nonlinearity, for the classical linear and nonlinear advection equations and for the advection reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the PDEs governing the response-excitation probability density function and we compare the numerical results against those obtained from probabilistic collocation (PCM) and multi-element probabilistic collocation (ME-PCM) methods. We have found that the response-excitation approach yields accurate representations of the statistical properties associated with the stochastic solution. The question of high-dimensionality for evolution equations involving multidimensional joint responseexcitation probability densities is also addressed. Suggested Reviewers: Anthony Nouy [email protected] Daniel Tartakovsky [email protected] Themistoklis Sapsis [email protected] Dear Editor: Please consider for publication in the Journal of Computational Physics the attached copy of the manuscript entitled ``New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs''. In this paper we determine new types of evolution equations satisfied by the joint response excitation probability density function associated with the stochastic solution to first-order nonlinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. This extends recent work on the subject, e.g., by J.-B. Chen and J. Li [Prob. Eng. Mech 24 (2009), pp. 51-59], that holds for ODEs with random parameters, to nonlinear PDEs. The new equations we obtain are of interest in many areas of mathematical physics since they can model, e.g., ocean waves in an Eulerian framework linear and nonlinear advection problems, advection-reaction systems and, more generally, scalar conservation laws. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the PDEs governing the response-excitation probability density function and we compare the numerical results against those obtained from probabilistic collocation (PCM) and multi-element probabilistic collocation (ME-PCM) methods. The numerical results suggest that the response excitation approach yields accurate representations of the statistical properties associated with the stochastic solution. Therefore, it can be effectively employed as a new computational method for uncertainty quantification. Thank you for your time and consideration. George EM Karniadakis Significance and Novelty of this paper