A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

• Published in: Journal of computational physics Vol. 230; no. 19; pp. 7332 - 7346
• Main Authors: Ren, Xiaoan, Wu, Wenquan, Xanthis, Leonidas S.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 10.08.2011; Elsevier
• Subjects: CALCULATION METHODS; CHAOS THEORY; Computation; Computational techniques; COMPUTERIZED SIMULATION; Dynamical systems; EQUATIONS; Exact sciences and technology; FUNCTIONS; MATHEMATICAL METHODS AND COMPUTING; Mathematical methods in physics; Mathematical models; MATHEMATICAL SOLUTIONS; MATHEMATICAL SPACE; MATHEMATICS; Physics; POLYNOMIALS; RANDOMNESS; RESOLUTION; SIMULATION; SPACE; Spectra; Spectral methods; Steep gradients in stochastic problems; Stochastic convection–diffusion; STOCHASTIC PROCESSES; Stochastic space-refinement; Stochastic wavelet multiscale solver; Stochasticity; Uncertainty; Uncertainty quantification; Wavelet; Uncertainty quantification; Stochastic space-refinement; Stochastic wavelet multiscale solver; Stochastic convection–diffusion; Steep gradients in stochastic problems; Spectral methods; Chaos; Uncertainty; Spectral method; Stochastic convection-diffusion; Adaptive method; Independent variable; Convection; Calculation methods; Wavelets; Deterministic system; Models; Calculation; Diffusion; Random variable
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2011.05.038

► New approach for stochastic computations based on polynomial chaos. ► Development of dynamically adaptive wavelet multiscale solver using space refinement. ► Accurate capture of steep gradients and multiscale features in stochastic problems. ► All scales of each random mode are captured on independent grids. ► Numerical examples demonstrate the need for different space resolutions per mode. In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.