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The moment closure problem is a fundamental challenge in fields such as gas dynamics, radiation transport, and biology. It involves estimating higher-order moments of an unknown probability distribution from lower-order moments. The difficulty lies in finding a way to close the moment equations that ensures they are hyperbolic and accurately capture transport dynamics. Despite various methods like Grad's closure, quadrature-based approaches, and entropy-based techniques, finding a globally hyperbolic, robust, and efficient solution remains an open question.

In our recent research, we propose a new approach to the moment closure problem based on orthogonal polynomials derived from Gram matrices, referred to as the "Gramian" closure. Its properties are studied in the context of the moment closure problem arising in gas kinetic theory, for which the proposed approach is proven to have multiple attractive mathematical properties. Numerical studies are carried out for model gas particle distributions and the approach is compared to other moment closure methods, such as Grad's closure and the maximum-entropy method. The proposed ``Gramian'' closure is shown to provide very accurate results for a wide range of distribution functions.

The figure shows relative error of the next higher moment for a range velocity shifts in the bimodal test case. The left part of the figure shows the relative error for even number of moments while the right one shows odd cases. Different plot markers with colors represent the closure. Note, that the Gramian and extended Gramian closure are defined in the even and odd case differently.

ArXiv Preprint

Contact: Eda Yilmaz