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Published in:
2018 | OriginalPaper | Chapter
A Stochastic Galerkin Method for the Fokker–Planck–Landau Equation with Random Uncertainties
Authors : Jingwei Hu, Shi Jin, Ruiwen Shu
Published in: Theory, Numerics and Applications of Hyperbolic Problems II
Publisher: Springer International Publishing
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Abstract
We propose a generalized polynomial chaos-based stochastic Galerkin method (gPC-sG) for the Fokker–Planck–Landau (FPL) equation with random uncertainties. The method can handle uncertainties from initial or boundary data and the neutralizing background. By a gPC expansion and the Galerkin projection, we convert the FPL equation with uncertainty into a system of deterministic equations. A consistency result is proven for the approximation of the collision operator. To compute efficiently the collision kernel under the gPC expansion, we use a singular value decomposition (SVD) combined with a fast spectral method for the collision operator. For high-dimensional random inputs, we adopt a sparse basis and use the sparsity of a set of basis-related coefficients and the Lax–Friedrichs splitting to avoid all the SVD involved. Numerical experiments verify the efficiency of the gPC-sG method.