TY - JOUR
T1 - An adaptive WENO collocation method for differential equations with random coefficients
AU - Guo, Wei
AU - Lin, Guang
AU - Christlieb, Andrew J.
AU - Qiu, Jingmei
N1 - Publisher Copyright:
© 2016 by the authors.
PY - 2016/6/1
Y1 - 2016/6/1
N2 - The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers' equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.
AB - The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers' equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.
KW - Adaptive mesh refinement
KW - High-order
KW - Multi-element
KW - Stochastic collocation method
KW - WENO interpolation
UR - http://www.scopus.com/inward/record.url?scp=85054239411&partnerID=8YFLogxK
U2 - 10.3390/math4020029
DO - 10.3390/math4020029
M3 - Article
AN - SCOPUS:85054239411
SN - 2227-7390
VL - 4
JO - Mathematics
JF - Mathematics
IS - 2
M1 - 29
ER -