On the quadratic stability of asymmetric Hermite basis with application to plasma physics with oscillating electric field

2024/05/13 by Ruiyang Dai, Bruno Després, Dai, Ruiyang +1 · 1 citation
Mathematics · Computer Science · Engineering · #Numerical methods for differential equations #Matrix Theory and Algorithms #Advanced Numerical Methods in Computational Mathematics

paper · pdf · doi:10.48550/arxiv.2405.07811

Abstract

We analyze why the discretization of linear transport with asymmetric Hermite basis functions can be instable in quadratic norm. The main reason is that the finite truncation of the infinite moment linear system looses the skew-symmetry property with respect to the Gram matrix. Then we propose an original closed formula for the scalar product of any pair of asymmetric basis functions. It makes possible the construction of two simple modifications of the linear systems which recover the skew-symmetry property. By construction the new methods are quadratically stable with respect to the natural L² norm. We explain how to generalize to other transport equations encountered in numerical plasma physics. Basic numerical tests with oscillating electric fields of different nature illustrate the unconditional stability properties of our algorithms.

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• Galerkin method for asymmetrically-weighted Hermite approximations applied to the Vlasov-Poisson system

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