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Articles written by Wan, X.

  1. Wan, X. and Karniadakis, G.E.. "An adaptive multi-element generalized Polynomial Chaos method for stochastic differential equations" J. Comput. Phys.. 209 (2). 2005. pp. 617--642.

    We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized polynomial chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan-Orszag three-mode problem, as well as advection-diffusion problems. The new method is similar to spectral element method for deterministic problems but with h-p discretization of the random space


  2. Wan, X. and Karniadakis, G.E.. "Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures" SIAM J. Sci. Comput.. 28 (3). 2006. pp. 901--928.

    We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities.


  3. Wan, X., Xiu, D., and Karniadakis, G.E.. "Stochastic Solutions for the Two-Dimensional Advection-Diffusion Equation" SIAM Journal on Scientific Computing. 26 (2). 2004. pp. 578--590.

    In this paper, we solve the two-dimensional advection-diffusion equation with random transport velocity. The generalized polynomial chaos expansion is employed to discretize the equation in random space while the spectral hp element method is used for spatial discretization. Numerical results which demonstrate the convergence of generalized polynomial chaos are presented. Specifically, it appears that the fast convergence rate in the variance is the same as that of the mean solution in the Jacobi-chaos unlike the Hermite-chaos. To this end, a new model to represent compact Gaussian distributions is also proposed.


  4. Wan, X. and Karniadakis, G.E.. "Beyond Wiener---Askey Expansions: Handling Arbitrary PDFs" Journal of Scientific Computing. 27 (1-3). 2006. pp. 455--464.

    In this paper we present a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with stochastic inputs with arbitrary probability measures. Based on the decomposition of the random space of the stochastic inputs, we construct numerically a set of orthogonal polynomials with respect to a conditional probability density function (PDF) in each element and subsequently implement generalized Polynomial Chaos (gPC) locally. Numerical examples show that ME-gPC exhibits both p- and h-convergence for arbitrary probability measures.


  5. Lin, G., Wan, X., Su, C.-H., and Karniadakis, G.E.. "Stochastic Computational Fluid Mechanics" Computing in Science and Engineering. 9 (2). 2007. pp. 21--29.


  6. Wan, X. and Karniadakis, G.E.. "Stochastic heat transfer enhancement in a grooved channel" Journal of Fluid Mechanics. vol. 565. OCT 2006. pp. 255--278.

    We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien-Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter E for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of E on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of E, which are skewed with their peaks increasing towards larger values of E as the Reynolds number increases. We then study the effect A multiple frequencies described by a periodically correlated random process. We find that the mean value of E is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing `smart' PDFs of the stochastic input that can result in improved heat transfer enhancement rates.


  7. Wan, X. and Karniadakis, G.E.. "Long-term behavior of polynomial chaos in stochastic flow simulations" Computer Methods in Applied Mechanics and Engineering. 195 (41-43). 2006. pp. 5582--5596.

    In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend these results to other random distributions and high-dimensional random inputs with numerical verification using the algebraic convergence rate of ME-gPC. Finally, we apply our results to noisy flow past a stationary circular cylinder. Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior. (c) 2005 Elsevier B.V. All rights reserved.