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Volume 30, Issue 2
A WENO-Based Stochastic Galerkin Scheme for Ideal MHD Equations with Random Inputs

Kailiang Wu, Dongbin Xiu & Xinghui Zhong

DOI: 10.4208/cicp.OA-2020-0167

Commun. Comput. Phys., 30 (2021), pp. 423-447.

Published online: 2021-05

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  • Abstract

In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochastic Galerkin approximation. A special treatment with symmetrization is carried out for the gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkin system is provably symmetric hyperbolic in the spatially one-dimensional case. We discretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended to two spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme.

  • Keywords

Uncertainty quantification, ideal magnetohydrodynamics, generalized polynomial chaos, stochastic Galerkin, symmetric hyperbolic, finite volume WENO method.

  • AMS Subject Headings

65M60, 65M08, 35L60, 76W05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-30-423, author = {Wu , KailiangXiu , Dongbin and Zhong , Xinghui}, title = {A WENO-Based Stochastic Galerkin Scheme for Ideal MHD Equations with Random Inputs}, journal = {Communications in Computational Physics}, year = {2021}, volume = {30}, number = {2}, pages = {423--447}, abstract = {

In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochastic Galerkin approximation. A special treatment with symmetrization is carried out for the gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkin system is provably symmetric hyperbolic in the spatially one-dimensional case. We discretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended to two spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2020-0167}, url = {http://global-sci.org/intro/article_detail/cicp/19120.html} }
TY - JOUR T1 - A WENO-Based Stochastic Galerkin Scheme for Ideal MHD Equations with Random Inputs AU - Wu , Kailiang AU - Xiu , Dongbin AU - Zhong , Xinghui JO - Communications in Computational Physics VL - 2 SP - 423 EP - 447 PY - 2021 DA - 2021/05 SN - 30 DO - http://doi.org/10.4208/cicp.OA-2020-0167 UR - https://global-sci.org/intro/article_detail/cicp/19120.html KW - Uncertainty quantification, ideal magnetohydrodynamics, generalized polynomial chaos, stochastic Galerkin, symmetric hyperbolic, finite volume WENO method. AB -

In this paper, we investigate the ideal magnetohydrodynamic (MHD) equations with random inputs based on generalized polynomial chaos (gPC) stochastic Galerkin approximation. A special treatment with symmetrization is carried out for the gPC stochastic Galerkin method so that the resulting deterministic gPC Galerkin system is provably symmetric hyperbolic in the spatially one-dimensional case. We discretize the hyperbolic gPC Galerkin system with a high-order path-conservative finite volume weighted essentially non-oscillatory scheme in space and a third-order total variation diminishing Runge-Kutta method in time. The method is also extended to two spatial dimensions via the operator splitting technique. Several numerical examples are provided to illustrate the accuracy and effectiveness of the numerical scheme.

Kailiang Wu, Dongbin Xiu & Xinghui Zhong. (2021). A WENO-Based Stochastic Galerkin Scheme for Ideal MHD Equations with Random Inputs. Communications in Computational Physics. 30 (2). 423-447. doi:10.4208/cicp.OA-2020-0167
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