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

• Published in: Journal of computational physics Vol. 231; no. 21; pp. 7450 - 7474
• Main Authors: Venturi, D., Karniadakis, G.E.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 30.08.2012; Elsevier
• Subjects: ADVECTION; BOUNDARY CONDITIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPARATIVE EVALUATIONS; Computational techniques; EVOLUTION; Exact sciences and technology; EXCITATION; Hopf characteristic functional; INTEGRALS; Mathematical analysis; Mathematical methods in physics; Mathematical models; NONLINEAR PROBLEMS; Nonlinearity; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; PDF methods; Physics; Probabilistic collocation; PROBABILISTIC ESTIMATION; Probabilistic methods; PROBABILITY DENSITY FUNCTIONS; Probability theory; RANDOMNESS; SCALARS; STOCHASTIC PROCESSES; Uncertainty quantification; Hopf characteristic functional; Uncertainty quantification; Probabilistic collocation; PDF methods; Dimensionality; Uncertainty; Spectral method; Partial differential equations; Nonlinear differential equations; Boundary conditions; Evolution equation; Calculation methods; Galerkin-Petrov method; Excitation functions; Advection; Initial conditions; Functionals; Numerical solution; First order; Probability density function; Calculation; Excitation; Forcing
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2012.07.013

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed 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 proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.