Local Galerkinʹs schemes qualitative analysis for 1D transport equation solving

Using the local projection method on the test function space associated to the elementary cell (.6.x,.6.J,L), the transport equation is reduced to a matrix equation. The method used here is called Green Matrix Method (GMM). For the SN approximation, the GMM formalism represents a generalization of the difference scheme without using the diamond relation. With GMM, the spatial integration presents truncation errors due to the consideration of a finite number of terms in the development of the angular flux in Legendre-Fourier series. For the SN approximation, on the base of the analytical expressions of the Green Matrix elements (GME), we qualitatively evaluate the GMM performances. Unlike the diamond difference scheme, the GMM algorithm presents a new type of truncation error for thin spatial meshes. This error is due to the approximate representation in rational fraction of the GME. Using symbolical methods, we also treat the two-cyclic iteration scheme for the S4 and DPI approximations. The two-cyclic algorithms developed both an improvement of precision and a substantial increase of the convergence rate comparatively to the classical iteration schema on the scattering source. Numerical tests present the dependence of the results precision on the spatial approximation order. The GMM is superior in point of precision comparatively to the diamond difference scheme