Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa's type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on ℙk discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order O(N -t + hk) for a smooth solution which admits partial derivatives of order k + 1 and t with respect to the spatial and angular variables, respectively. For k = 0 (piecewise constant finite elements) we also obtain a convergence result of order O (N - t + h 1 2). Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the L2-norm, typically of one more order, O (N - t + h k + 1).